Theorem iffalsei 4486

Description: Inference associated with iffalse 4485. (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 4485	. 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 4476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-if 4477

This theorem is referenced by:  ssttrcl  9616  ttrclselem2  9627  sum0  15635  prod0  15857  prmo4  17046  prmo6  17048  itg0  25728  vieta1lem2  26266  right1s  27861  vtxval0  29038  iedgval0  29039  ex-prmo  30460  dfrdg2  35909  dfrdg4  36067  fwddifnp1  36281  bj-pr21val  37130  bj-pr22val  37136  imsqrtvalex  43803  clsk1indlem4  44201  clsk1indlem1  44202  refsum2cnlem1  45198  limsup10ex  45933  iblempty  46125  fouriersw  46391