Theorem iffalsei 4477

Description: Inference associated with iffalse 4476. (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 4476	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1537  ifcif 4467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-if 4468

This theorem is referenced by:  sum0  15078  prod0  15297  prmo4  16461  prmo6  16463  itg0  24380  vieta1lem2  24900  vtxval0  26824  iedgval0  26825  ex-prmo  28238  dfrdg2  33040  dfrdg4  33412  fwddifnp1  33626  bj-pr21val  34328  bj-pr22val  34334  clsk1indlem4  40414  clsk1indlem1  40415  refsum2cnlem1  41314  limsup10ex  42074  iblempty  42270  fouriersw  42536