Theorem elimhyps2 38476

Description: Generalization of elimhyps 38473 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)

Hypothesis

Ref	Expression
elimhyps2.1	⊢ [𝐵 / 𝑥]𝜑

Assertion

Ref	Expression
elimhyps2	⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Proof of Theorem elimhyps2

Step	Hyp	Ref	Expression
1		dfsbcq 3780	. 2 ⊢ (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
2		dfsbcq 3780	. 2 ⊢ (𝐵 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐵 / 𝑥]𝜑 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
3		elimhyps2.1	. 2 ⊢ [𝐵 / 𝑥]𝜑
4	1, 2, 3	elimhyp 4597	1 ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Syntax hints:  [wsbc 3778  ifcif 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-sbc 3779  df-if 4533

This theorem is referenced by: (None)