Theorem elimhyps 39597

Description: A version of elimhyp 4549 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 3758	. 2 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2		dfsbcq 3749	. 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 4549	1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Syntax hints:  [wsbc 3747  ifcif 4483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-sbc 3748  df-if 4484

This theorem is referenced by:  renegclALT  39599