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 1570  ifcif 4488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-if 4489

This theorem is referenced by:  ssttrcl  9685  ttrclselem2  9696  sum0  15774  prod0  15999  prmo4  17189  prmo6  17191  itg0  25920  vieta1lem2  26453  right1s  28070  vtxval0  29370  iedgval0  29371  ex-prmo  30791  dfrdg2  36266  dfrdg4  36424  fwddifnp1  36638  bj-pr21val  37630  bj-pr22val  37636  imsqrtvalex  44355  clsk1indlem4  44753  clsk1indlem1  44754  refsum2cnlem1  45740  limsup10ex  46470  iblempty  46662  fouriersw  46928