Theorem iffalsei 4510

Description: Inference associated with iffalse 4509. (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 4509	. 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 4500

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-if 4501

This theorem is referenced by:  ssttrcl  9729  ttrclselem2  9740  sum0  15737  prod0  15959  prmo4  17147  prmo6  17149  itg0  25733  vieta1lem2  26271  right1s  27859  vtxval0  29018  iedgval0  29019  ex-prmo  30440  dfrdg2  35813  dfrdg4  35969  fwddifnp1  36183  bj-pr21val  37031  bj-pr22val  37037  imsqrtvalex  43670  clsk1indlem4  44068  clsk1indlem1  44069  refsum2cnlem1  45061  limsup10ex  45802  iblempty  45994  fouriersw  46260