Theorem iffalsei 4489

Description: Inference associated with iffalse 4488. (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 4488	. 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 4479

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-if 4480

This theorem is referenced by:  ssttrcl  9624  ttrclselem2  9635  sum0  15644  prod0  15866  prmo4  17055  prmo6  17057  itg0  25737  vieta1lem2  26275  right1s  27892  vtxval0  29112  iedgval0  29113  ex-prmo  30534  dfrdg2  35987  dfrdg4  36145  fwddifnp1  36359  bj-pr21val  37214  bj-pr22val  37220  imsqrtvalex  43887  clsk1indlem4  44285  clsk1indlem1  44286  refsum2cnlem1  45282  limsup10ex  46017  iblempty  46209  fouriersw  46475