Theorem iffalsei 4558

Description: Inference associated with iffalse 4557. (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 4557	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1537  ifcif 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-if 4549

This theorem is referenced by:  ssttrcl  9784  ttrclselem2  9795  sum0  15769  prod0  15991  prmo4  17175  prmo6  17177  itg0  25835  vieta1lem2  26371  right1s  27952  vtxval0  29074  iedgval0  29075  ex-prmo  30491  dfrdg2  35759  dfrdg4  35915  fwddifnp1  36129  bj-pr21val  36979  bj-pr22val  36985  imsqrtvalex  43608  clsk1indlem4  44006  clsk1indlem1  44007  refsum2cnlem1  44937  limsup10ex  45694  iblempty  45886  fouriersw  46152