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Theorem elimhyps 36089
Description: A version of elimhyp 4528 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
Hypothesis
Ref Expression
elimhyps.1 [𝐵 / 𝑥]𝜑
Assertion
Ref Expression
elimhyps [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Proof of Theorem elimhyps
StepHypRef Expression
1 sbceq1a 3781 . 2 (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2 dfsbcq 3772 . 2 (𝐵 = if(𝜑, 𝑥, 𝐵) → ([𝐵 / 𝑥]𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
3 elimhyps.1 . 2 [𝐵 / 𝑥]𝜑
41, 2, 3elimhyp 4528 1 [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  [wsbc 3770  ifcif 4465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-sbc 3771  df-if 4466
This theorem is referenced by:  renegclALT  36091
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