Theorem iffalsei 4501

Description: Inference associated with iffalse 4500. (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 4500	. 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 4491

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-if 4492

This theorem is referenced by:  ssttrcl  9675  ttrclselem2  9686  sum0  15694  prod0  15916  prmo4  17105  prmo6  17107  itg0  25688  vieta1lem2  26226  right1s  27814  vtxval0  28973  iedgval0  28974  ex-prmo  30395  dfrdg2  35790  dfrdg4  35946  fwddifnp1  36160  bj-pr21val  37008  bj-pr22val  37014  imsqrtvalex  43642  clsk1indlem4  44040  clsk1indlem1  44041  refsum2cnlem1  45038  limsup10ex  45778  iblempty  45970  fouriersw  46236