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Theorem elimhyps 38943
Description: A version of elimhyp 4596 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 3802 . 2 (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2 dfsbcq 3793 . 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 4596 1 [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  [wsbc 3791  ifcif 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792  df-if 4532
This theorem is referenced by:  renegclALT  38945
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