Theorem iffalsei 4480

Description: Inference associated with iffalse 4479. (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 4479	. 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 4470

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-if 4471

This theorem is referenced by:  ssttrcl  9600  ttrclselem2  9611  sum0  15623  prod0  15845  prmo4  17034  prmo6  17036  itg0  25703  vieta1lem2  26241  right1s  27836  vtxval0  29012  iedgval0  29013  ex-prmo  30431  dfrdg2  35829  dfrdg4  35985  fwddifnp1  36199  bj-pr21val  37047  bj-pr22val  37053  imsqrtvalex  43679  clsk1indlem4  44077  clsk1indlem1  44078  refsum2cnlem1  45074  limsup10ex  45811  iblempty  46003  fouriersw  46269