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Theorem iftruei 3542
Description: Inference associated with iftrue 3541. (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 3541 . 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 3536
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 3537
This theorem is referenced by:  ctmlemr  7109  xnegpnf  9830  xnegmnf  9831  xaddpnf1  9848  xaddpnf2  9849  xaddmnf1  9850  xaddmnf2  9851  pnfaddmnf  9852  mnfaddpnf  9853  iseqf1olemqk  10496  exp0  10526  sumsnf  11419  prodsnf  11602  lcm0val  12067  ennnfonelemj0  12404  ennnfonelem0  12408  mulg0  12993  lgs0  14453  lgs2  14457  peano3nninf  14795  dceqnconst  14847
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