Theorem iffalsei 4474

Description: Inference associated with iffalse 4473. (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 4473	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1541  ifcif 4464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-if 4465

This theorem is referenced by:  ssttrcl  9434  ttrclselem2  9445  sum0  15414  prod0  15634  prmo4  16810  prmo6  16812  itg0  24925  vieta1lem2  25452  vtxval0  27390  iedgval0  27391  ex-prmo  28802  dfrdg2  33750  dfrdg4  34232  fwddifnp1  34446  bj-pr21val  35182  bj-pr22val  35188  imsqrtvalex  41207  clsk1indlem4  41607  clsk1indlem1  41608  refsum2cnlem1  42533  limsup10ex  43268  iblempty  43460  fouriersw  43726