Theorem iftruei 3531

Description: Inference associated with iftrue 3530. (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 3530	. 2 |- ( ph -> if ( ph , A , B ) = A )
3	1, 2	ax-mp 5	1 |- if ( ph , A , B ) = A

Syntax hints:   = wceq 1348   ifcif 3525

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 3526

This theorem is referenced by:  ctmlemr  7081  xnegpnf  9772  xnegmnf  9773  xaddpnf1  9790  xaddpnf2  9791  xaddmnf1  9792  xaddmnf2  9793  pnfaddmnf  9794  mnfaddpnf  9795  iseqf1olemqk  10437  exp0  10467  sumsnf  11359  prodsnf  11542  lcm0val  12006  ennnfonelemj0  12343  ennnfonelem0  12347  lgs0  13629  lgs2  13633  peano3nninf  13962  dceqnconst  14013