Theorem iffalsei 4535

Description: Inference associated with iffalse 4534. (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 4534	. 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 4525

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4526

This theorem is referenced by:  ssttrcl  9755  ttrclselem2  9766  sum0  15757  prod0  15979  prmo4  17165  prmo6  17167  itg0  25815  vieta1lem2  26353  right1s  27934  vtxval0  29056  iedgval0  29057  ex-prmo  30478  dfrdg2  35796  dfrdg4  35952  fwddifnp1  36166  bj-pr21val  37014  bj-pr22val  37020  imsqrtvalex  43659  clsk1indlem4  44057  clsk1indlem1  44058  refsum2cnlem1  45042  limsup10ex  45788  iblempty  45980  fouriersw  46246