Theorem iftruei 3405

Description: Inference associated with iftrue 3404. (Contributed by BJ, 7-Oct-2018.)

Hypothesis

Ref	Expression
iftruei.1	|- ph

Assertion

Ref	Expression
iftruei	|- if ( ph , A , B ) = A

Proof of Theorem iftruei

Step	Hyp	Ref	Expression
1		iftruei.1	. 2 |- ph
2		iftrue 3404	. 2 |- ( ph -> if ( ph , A , B ) = A )
3	1, 2	ax-mp 7	1 |- if ( ph , A , B ) = A

Syntax hints:   = wceq 1290   ifcif 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-if 3400

This theorem is referenced by:  ctmlemr  6846  xnegpnf  9353  xnegmnf  9354  iseqf1olemqk  9986  exp0  10022  sumsnf  10866  lcm0val  11388  peano3nninf  12200