Theorem iffalsei 4495

Description: Inference associated with iffalse 4494. (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 4494	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-if 4486

This theorem is used by:  ssttrcl  9753  ttrclselem2  9764  sum0  15931  prod0  16154  prmo4  17350  prmo6  17352  itg0  26155  vieta1lem2  26689  right1s  28339  vtxval0  29674  iedgval0  29675  ex-prmo  31117  dfrdg2  36772  dfrdg4  36930  fwddifnp1  37145  bj-pr21val  38141  bj-pr22val  38147  imsqrtvalex  44907  clsk1indlem4  45305  clsk1indlem1  45306  refsum2cnlem1  46328  limsup10ex  47056  iblempty  47248  fouriersw  47514  parswapprop  51108  par2hom  51110