Theorem iffalsei 4354

Description: Inference associated with iffalse 4353. (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 4353	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1508  ifcif 4344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-if 4345

This theorem is referenced by:  sum0  14936  prod0  15155  prmo4  16315  prmo6  16317  itg0  24098  vieta1lem2  24618  vtxval0  26542  iedgval0  26543  ex-prmo  28031  dfrdg2  32598  dfrdg4  32970  fwddifnp1  33184  bj-pr21val  33880  bj-pr22val  33886  clsk1indlem4  39795  clsk1indlem1  39796  refsum2cnlem1  40751  limsup10ex  41519  iblempty  41714  fouriersw  41981