Theorem iffalsei 4491

Description: Inference associated with iffalse 4490. (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 4490	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1542  ifcif 4481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-if 4482

This theorem is referenced by:  ssttrcl  9636  ttrclselem2  9647  sum0  15656  prod0  15878  prmo4  17067  prmo6  17069  itg0  25749  vieta1lem2  26287  right1s  27904  vtxval0  29124  iedgval0  29125  ex-prmo  30546  dfrdg2  36006  dfrdg4  36164  fwddifnp1  36378  bj-pr21val  37258  bj-pr22val  37264  imsqrtvalex  43999  clsk1indlem4  44397  clsk1indlem1  44398  refsum2cnlem1  45394  limsup10ex  46128  iblempty  46320  fouriersw  46586