Theorem iffalsei 4435

Description: Inference associated with iffalse 4434. (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 4434	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1543  ifcif 4425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-if 4426

This theorem is referenced by:  sum0  15250  prod0  15468  prmo4  16644  prmo6  16646  itg0  24631  vieta1lem2  25158  vtxval0  27084  iedgval0  27085  ex-prmo  28496  dfrdg2  33441  dfrdg4  33939  fwddifnp1  34153  bj-pr21val  34889  bj-pr22val  34895  imsqrtvalex  40871  clsk1indlem4  41272  clsk1indlem1  41273  refsum2cnlem1  42194  limsup10ex  42932  iblempty  43124  fouriersw  43390