Theorem iffalsei 4539

Description: Inference associated with iffalse 4538. (Contributed by BJ, 7-Oct-2018.)

Hypothesis

Ref	Expression
iffalsei.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
iffalsei	⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Proof of Theorem iffalsei

Step	Hyp	Ref	Expression
1		iffalsei.1	. 2 ⊢ ¬ 𝜑
2		iffalse 4538	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1534  ifcif 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-if 4530

This theorem is referenced by:  ssttrcl  9739  ttrclselem2  9750  sum0  15700  prod0  15920  prmo4  17097  prmo6  17099  itg0  25722  vieta1lem2  26259  right1s  27835  vtxval0  28865  iedgval0  28866  ex-prmo  30282  dfrdg2  35391  dfrdg4  35547  fwddifnp1  35761  bj-pr21val  36492  bj-pr22val  36498  imsqrtvalex  43076  clsk1indlem4  43474  clsk1indlem1  43475  refsum2cnlem1  44399  limsup10ex  45161  iblempty  45353  fouriersw  45619