Theorem iffalsei 4464

Description: Inference associated with iffalse 4463. (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 4463	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1547  ifcif 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-if 4455

This theorem is referenced by:  ssttrcl  9627  ttrclselem2  9638  sum0  15674  prod0  15899  prmo4  17089  prmo6  17091  itg0  25765  vieta1lem2  26295  right1s  27906  vtxval0  29126  iedgval0  29127  ex-prmo  30547  dfrdg2  36021  dfrdg4  36179  fwddifnp1  36393  bj-pr21val  37366  bj-pr22val  37372  imsqrtvalex  44090  clsk1indlem4  44488  clsk1indlem1  44489  refsum2cnlem1  45485  limsup10ex  46216  iblempty  46408  fouriersw  46674