Theorem iftruei 3419

Description: Inference associated with iftrue 3418. (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

Step	Hyp	Ref	Expression
1		iftruei.1	. 2 ⊢ 𝜑
2		iftrue 3418	. 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	1, 2	ax-mp 7	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴

Syntax hints:   = wceq 1296  ifcif 3413

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 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-if 3414

This theorem is referenced by:  ctmlemr  6870  xnegpnf  9394  xnegmnf  9395  xaddpnf1  9412  xaddpnf2  9413  xaddmnf1  9414  xaddmnf2  9415  pnfaddmnf  9416  mnfaddpnf  9417  iseqf1olemqk  10044  exp0  10074  sumsnf  10952  lcm0val  11474  peano3nninf  12602