Theorem iffalsei 4541

Description: Inference associated with iffalse 4540. (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 4540	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1537  ifcif 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-if 4532

This theorem is referenced by:  ssttrcl  9753  ttrclselem2  9764  sum0  15754  prod0  15976  prmo4  17162  prmo6  17164  itg0  25830  vieta1lem2  26368  right1s  27949  vtxval0  29071  iedgval0  29072  ex-prmo  30488  dfrdg2  35777  dfrdg4  35933  fwddifnp1  36147  bj-pr21val  36996  bj-pr22val  37002  imsqrtvalex  43636  clsk1indlem4  44034  clsk1indlem1  44035  refsum2cnlem1  44975  limsup10ex  45729  iblempty  45921  fouriersw  46187