Theorem iffalsei 4534

Description: Inference associated with iffalse 4533. (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 4533	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1542  ifcif 4524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-if 4525

This theorem is referenced by:  ssttrcl  9697  ttrclselem2  9708  sum0  15654  prod0  15874  prmo4  17048  prmo6  17050  itg0  25266  vieta1lem2  25793  right1s  27357  vtxval0  28266  iedgval0  28267  ex-prmo  29679  dfrdg2  34698  dfrdg4  34854  fwddifnp1  35068  bj-pr21val  35799  bj-pr22val  35805  imsqrtvalex  42268  clsk1indlem4  42666  clsk1indlem1  42667  refsum2cnlem1  43592  limsup10ex  44362  iblempty  44554  fouriersw  44820