Theorem iffalsei 4475

Description: Inference associated with iffalse 4474. (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 4474	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1539  ifcif 4465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-if 4466

This theorem is referenced by:  ssttrcl  9521  ttrclselem2  9532  sum0  15482  prod0  15702  prmo4  16878  prmo6  16880  itg0  24993  vieta1lem2  25520  vtxval0  27458  iedgval0  27459  ex-prmo  28872  dfrdg2  33820  dfrdg4  34302  fwddifnp1  34516  bj-pr21val  35251  bj-pr22val  35257  imsqrtvalex  41467  clsk1indlem4  41867  clsk1indlem1  41868  refsum2cnlem1  42793  limsup10ex  43543  iblempty  43735  fouriersw  44001