Theorem iffalsei 4487

Description: Inference associated with iffalse 4486. (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 4486	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1559  ifcif 4477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4478

This theorem is referenced by:  ssttrcl  9664  ttrclselem2  9675  sum0  15739  prod0  15964  prmo4  17155  prmo6  17157  itg0  25830  vieta1lem2  26363  right1s  27977  vtxval0  29197  iedgval0  29198  ex-prmo  30618  dfrdg2  36104  dfrdg4  36262  fwddifnp1  36476  bj-pr21val  37459  bj-pr22val  37465  imsqrtvalex  44183  clsk1indlem4  44581  clsk1indlem1  44582  refsum2cnlem1  45578  limsup10ex  46308  iblempty  46500  fouriersw  46766