Theorem iffalsei 4488

Description: Inference associated with iffalse 4487. (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 4487	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1540  ifcif 4478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-if 4479

This theorem is referenced by:  ssttrcl  9630  ttrclselem2  9641  sum0  15646  prod0  15868  prmo4  17057  prmo6  17059  itg0  25697  vieta1lem2  26235  right1s  27828  vtxval0  29002  iedgval0  29003  ex-prmo  30421  dfrdg2  35771  dfrdg4  35927  fwddifnp1  36141  bj-pr21val  36989  bj-pr22val  36995  imsqrtvalex  43622  clsk1indlem4  44020  clsk1indlem1  44021  refsum2cnlem1  45018  limsup10ex  45758  iblempty  45950  fouriersw  46216