Theorem sbcie2g 3783

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3784 avoids a disjointness condition on 𝑥, 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
sbcie2g.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
sbcie2g.2	⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbcie2g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜒,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem sbcie2g

Step	Hyp	Ref	Expression
1		dfsbcq 3744	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		sbcie2g.2	. 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
3		sbsbc 3746	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4		sbcie2g.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	sbievw 2099	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	3, 5	bitr3i 277	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
7	1, 2, 6	vtoclbg 3516	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  [wsb 2068   ∈ wcel 2114  [wsbc 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3743

This theorem is referenced by:  sbcel2gv  3809  csbie2g  3891  brab1  5148  bnj90  34898  bnj124  35046  riotasvd  39329