Theorem iffalsei 4492

Description: Inference associated with iffalse 4491. (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 4491	. 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 1570  ifcif 4482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-if 4483

This theorem is used by:  ssttrcl  9694  ttrclselem2  9705  sum0  15807  prod0  16030  prmo4  17220  prmo6  17222  itg0  26007  vieta1lem2  26543  right1s  28161  vtxval0  29496  iedgval0  29497  ex-prmo  30939  dfrdg2  36372  dfrdg4  36530  fwddifnp1  36745  bj-pr21val  37757  bj-pr22val  37763  imsqrtvalex  44486  clsk1indlem4  44884  clsk1indlem1  44885  refsum2cnlem1  45871  limsup10ex  46601  iblempty  46793  fouriersw  47059