Theorem iffalsei 4530

Description: Inference associated with iffalse 4529. (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 4529	. 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 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-if 4521

This theorem is referenced by:  ssttrcl  9705  ttrclselem2  9716  sum0  15663  prod0  15883  prmo4  17057  prmo6  17059  itg0  25619  vieta1lem2  26153  right1s  27726  vtxval0  28723  iedgval0  28724  ex-prmo  30136  dfrdg2  35228  dfrdg4  35384  fwddifnp1  35598  bj-pr21val  36350  bj-pr22val  36356  imsqrtvalex  42852  clsk1indlem4  43250  clsk1indlem1  43251  refsum2cnlem1  44176  limsup10ex  44940  iblempty  45132  fouriersw  45398