Theorem iffalsei 4499

Description: Inference associated with iffalse 4498. (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 4498	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1567  ifcif 4489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-if 4490

This theorem is referenced by:  ssttrcl  9680  ttrclselem2  9691  sum0  15768  prod0  15993  prmo4  17184  prmo6  17186  itg0  25904  vieta1lem2  26437  right1s  28051  vtxval0  29326  iedgval0  29327  ex-prmo  30747  dfrdg2  36180  dfrdg4  36338  fwddifnp1  36552  bj-pr21val  37533  bj-pr22val  37539  imsqrtvalex  44257  clsk1indlem4  44655  clsk1indlem1  44656  refsum2cnlem1  45642  limsup10ex  46372  iblempty  46564  fouriersw  46830