Theorem iffalsei 4491

Description: Inference associated with iffalse 4490. (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 4490	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569  ifcif 4481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-if 4482

This theorem is used by:  ssttrcl  9720  ttrclselem2  9731  sum0  15893  prod0  16116  prmo4  17312  prmo6  17314  itg0  26106  vieta1lem2  26640  right1s  28288  vtxval0  29623  iedgval0  29624  ex-prmo  31066  dfrdg2  36650  dfrdg4  36808  fwddifnp1  37023  bj-pr21val  38019  bj-pr22val  38025  imsqrtvalex  44740  clsk1indlem4  45138  clsk1indlem1  45139  refsum2cnlem1  46148  limsup10ex  46877  iblempty  47069  fouriersw  47335  parswapprop  50871  par2hom  50873