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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-if 4486

This theorem is used by:  ssttrcl  9698  ttrclselem2  9709  sum0  15811  prod0  16036  prmo4  17226  prmo6  17228  itg0  26014  vieta1lem2  26550  right1s  28169  vtxval0  29504  iedgval0  29505  ex-prmo  30947  dfrdg2  36380  dfrdg4  36538  fwddifnp1  36753  bj-pr21val  37765  bj-pr22val  37771  imsqrtvalex  44494  clsk1indlem4  44892  clsk1indlem1  44893  refsum2cnlem1  45879  limsup10ex  46609  iblempty  46801  fouriersw  47067