Theorem sbcie2g 3598

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3599 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 3566	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		sbcie2g.2	. 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
3		sbsbc 3568	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4		nfv 1980	. . . 4 ⊢ Ⅎ𝑥𝜓
5		sbcie2g.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	4, 5	sbie 2533	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
7	3, 6	bitr3i 266	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
8	1, 2, 7	vtoclbg 3395	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1620  [wsb 2034   ∈ wcel 2127  [wsbc 3564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-12 2184  ax-13 2379  ax-ext 2728

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-v 3330  df-sbc 3565

This theorem is referenced by:  sbcel2gv  3625  csbie2g  3693  brab1  4840  bnj90  31068  bnj124  31219  riotasvd  34714