Theorem elimhyps 38917

Description: A version of elimhyp 4613 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

Step	Hyp	Ref	Expression
1		sbceq1a 3815	. 2 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2		dfsbcq 3806	. 2 ⊢ (𝐵 = if(𝜑, 𝑥, 𝐵) → ([𝐵 / 𝑥]𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
3		elimhyps.1	. 2 ⊢ [𝐵 / 𝑥]𝜑
4	1, 2, 3	elimhyp 4613	1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Syntax hints:  [wsbc 3804  ifcif 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805  df-if 4549

This theorem is referenced by:  renegclALT  38919