Theorem sbc2iegf 3016

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 108	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
2		simpl 108	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊)
3		sbc2iegf.4	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
4	3	adantll 468	. . . 4 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
5		nfv 1515	. . . 4 ⊢ Ⅎ𝑦(𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴)
6		sbc2iegf.2	. . . . 5 ⊢ Ⅎ𝑦𝜓
7	6	a1i 9	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜓)
8	2, 4, 5, 7	sbciedf 2981	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓))
9	8	adantll 468	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓))
10		nfv 1515	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
11		sbc2iegf.3	. . 3 ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊
12	10, 11	nfan 1552	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)
13		sbc2iegf.1	. . 3 ⊢ Ⅎ𝑥𝜓
14	13	a1i 9	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Ⅎ𝑥𝜓)
15	1, 9, 12, 14	sbciedf 2981	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1342  Ⅎwnf 1447   ∈ wcel 2135  [wsbc 2946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-sbc 2947

This theorem is referenced by:  sbc2ie  3017  opelopabaf  4245