Theorem sbc2iedv 3811

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Hypotheses

Ref	Expression
sbc2iedv.1	⊢ 𝐴 ∈ V
sbc2iedv.2	⊢ 𝐵 ∈ V
sbc2iedv.3	⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
sbc2iedv	⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

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

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2iedv

Step	Hyp	Ref	Expression
1		sbc2iedv.1	. . 3 ⊢ 𝐴 ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		sbc2iedv.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
5		sbc2iedv.3	. . . 4 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))
6	5	impl 456	. . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
7	4, 6	sbcied 3771	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
8	2, 7	sbcied 3771	1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441  [wsbc 3726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-sbc 3727

This theorem is referenced by:  dfoprab3  7954  sbcie2s  16951  ismnddef  18476  isfrgr  28853  sdclem1  35999