Theorem iffalsei 4538

Description: Inference associated with iffalse 4537. (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 4537	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1542  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-if 4529

This theorem is referenced by:  ssttrcl  9707  ttrclselem2  9718  sum0  15664  prod0  15884  prmo4  17058  prmo6  17060  itg0  25289  vieta1lem2  25816  right1s  27380  vtxval0  28289  iedgval0  28290  ex-prmo  29702  dfrdg2  34756  dfrdg4  34912  fwddifnp1  35126  bj-pr21val  35883  bj-pr22val  35889  imsqrtvalex  42383  clsk1indlem4  42781  clsk1indlem1  42782  refsum2cnlem1  43707  limsup10ex  44476  iblempty  44668  fouriersw  44934