Theorem iffalsei 4476

Description: Inference associated with iffalse 4475. (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 4475	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1533  ifcif 4466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-if 4467

This theorem is referenced by:  sum0  15072  prod0  15291  prmo4  16455  prmo6  16457  itg0  24374  vieta1lem2  24894  vtxval0  26818  iedgval0  26819  ex-prmo  28232  dfrdg2  33035  dfrdg4  33407  fwddifnp1  33621  bj-pr21val  34320  bj-pr22val  34326  clsk1indlem4  40387  clsk1indlem1  40388  refsum2cnlem1  41287  limsup10ex  42047  iblempty  42243  fouriersw  42510