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Theorem iftruei 3540
Description: Inference associated with iftrue 3539. (Contributed by BJ, 7-Oct-2018.)
Hypothesis
Ref Expression
iftruei.1 𝜑
Assertion
Ref Expression
iftruei if(𝜑, 𝐴, 𝐵) = 𝐴

Proof of Theorem iftruei
StepHypRef Expression
1 iftruei.1 . 2 𝜑
2 iftrue 3539 . 2 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
31, 2ax-mp 5 1 if(𝜑, 𝐴, 𝐵) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1353  ifcif 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3535
This theorem is referenced by:  ctmlemr  7106  xnegpnf  9826  xnegmnf  9827  xaddpnf1  9844  xaddpnf2  9845  xaddmnf1  9846  xaddmnf2  9847  pnfaddmnf  9848  mnfaddpnf  9849  iseqf1olemqk  10491  exp0  10521  sumsnf  11412  prodsnf  11595  lcm0val  12059  ennnfonelemj0  12396  ennnfonelem0  12400  mulg0  12942  lgs0  14307  lgs2  14311  peano3nninf  14638  dceqnconst  14689
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