Theorem iffalsei 4498

Description: Inference associated with iffalse 4497. (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 4497	. 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 4488

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 4489

This theorem is referenced by:  ssttrcl  9668  ttrclselem2  9679  sum0  15687  prod0  15909  prmo4  17098  prmo6  17100  itg0  25681  vieta1lem2  26219  right1s  27807  vtxval0  28966  iedgval0  28967  ex-prmo  30388  dfrdg2  35783  dfrdg4  35939  fwddifnp1  36153  bj-pr21val  37001  bj-pr22val  37007  imsqrtvalex  43635  clsk1indlem4  44033  clsk1indlem1  44034  refsum2cnlem1  45031  limsup10ex  45771  iblempty  45963  fouriersw  46229