Theorem elimhyps 40111

Description: A version of elimhyp 4547 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 3749	. 2 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2		dfsbcq 3740	. 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 4547	1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

This proof depends on syntax axioms:  [wsbc 3738  ifcif 4481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3739  df-if 4482

This theorem is used by:  renegclALT  40113