Theorem elimhyps 33265

Description: A version of elimhyp 4096 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 3413	. . 3 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2		dfsbcq 3404	. . 3 ⊢ (𝐵 = if(𝜑, 𝑥, 𝐵) → ([𝐵 / 𝑥]𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
3		elimhyps.1	. . 3 ⊢ [𝐵 / 𝑥]𝜑
4	1, 2, 3	elimhyp 4096	. 2 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
5		biid 250	. . 3 ⊢ (𝜑 ↔ 𝜑)
6		ifbi 4057	. . 3 ⊢ ((𝜑 ↔ 𝜑) → if(𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
7		dfsbcq 3404	. . . 4 ⊢ (if(𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
8	7	bicomd 212	. . 3 ⊢ (if(𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
9	5, 6, 8	mp2b 10	. 2 ⊢ ([if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑)
10	4, 9	mpbir 220	1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Syntax hints:   ↔ wb 195   = wceq 1475  [wsbc 3402  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403  df-if 4037

This theorem is referenced by:  renegclALT  33267