Theorem iftruei 3532

Description: Inference associated with iftrue 3531. (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 3531	. 2 |- ( ph -> if ( ph , A , B ) = A )
3	1, 2	ax-mp 5	1 |- if ( ph , A , B ) = A

Syntax hints:   = wceq 1348   ifcif 3526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-if 3527

This theorem is referenced by:  ctmlemr  7085  xnegpnf  9785  xnegmnf  9786  xaddpnf1  9803  xaddpnf2  9804  xaddmnf1  9805  xaddmnf2  9806  pnfaddmnf  9807  mnfaddpnf  9808  iseqf1olemqk  10450  exp0  10480  sumsnf  11372  prodsnf  11555  lcm0val  12019  ennnfonelemj0  12356  ennnfonelem0  12360  lgs0  13708  lgs2  13712  peano3nninf  14040  dceqnconst  14091