Theorem elimhyps 38135

Description: A version of elimhyp 4593 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 3788	. 2 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2		dfsbcq 3779	. 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 4593	1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Syntax hints:  [wsbc 3777  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-sbc 3778  df-if 4529

This theorem is referenced by:  renegclALT  38137