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

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ifcif 4489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-if 4490

This theorem is used by:  ssttrcl  9687  ttrclselem2  9698  sum0  15790  prod0  16015  prmo4  17205  prmo6  17207  itg0  25968  vieta1lem2  26501  right1s  28118  vtxval0  29418  iedgval0  29419  ex-prmo  30839  dfrdg2  36298  dfrdg4  36456  fwddifnp1  36670  bj-pr21val  37682  bj-pr22val  37688  imsqrtvalex  44405  clsk1indlem4  44803  clsk1indlem1  44804  refsum2cnlem1  45790  limsup10ex  46520  iblempty  46712  fouriersw  46978