Theorem sbc2iegf 3795

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
sbc2iegf.1	⊢ Ⅎ𝑥𝜓
sbc2iegf.2	⊢ Ⅎ𝑦𝜓
sbc2iegf.3	⊢ Ⅎ𝑥 𝐵 ∈ 𝑊
sbc2iegf.4	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbc2iegf	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝑉   𝑦,𝑊

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐵(𝑥)   𝑉(𝑦)   𝑊(𝑥)

Proof of Theorem sbc2iegf

Step	Hyp	Ref	Expression
1		simpl 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
2		simpl 486	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊)
3		sbc2iegf.4	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
4	3	adantll 713	. . . 4 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
5		nfv 1915	. . . 4 ⊢ Ⅎ𝑦(𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴)
6		sbc2iegf.2	. . . . 5 ⊢ Ⅎ𝑦𝜓
7	6	a1i 11	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜓)
8	2, 4, 5, 7	sbciedf 3761	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓))
9	8	adantll 713	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓))
10		nfv 1915	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
11		sbc2iegf.3	. . 3 ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊
12	10, 11	nfan 1900	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)
13		sbc2iegf.1	. . 3 ⊢ Ⅎ𝑥𝜓
14	13	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Ⅎ𝑥𝜓)
15	1, 9, 12, 14	sbciedf 3761	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  [wsbc 3720

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721

This theorem is referenced by:  sbc2ie  3796  vtocl2dOLD  3876  opelopabaf  5396  elmptrab  22432