Theorem sbc2iegf 3113

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 443	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → A ∈ V)
2		simpl 443	. . . 4 ⊢ ((B ∈ W ∧ x = A) → B ∈ W)
3		sbc2iegf.4	. . . . 5 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
4	3	adantll 694	. . . 4 ⊢ (((B ∈ W ∧ x = A) ∧ y = B) → (φ ↔ ψ))
5		nfv 1619	. . . 4 ⊢ Ⅎy(B ∈ W ∧ x = A)
6		sbc2iegf.2	. . . . 5 ⊢ Ⅎyψ
7	6	a1i 10	. . . 4 ⊢ ((B ∈ W ∧ x = A) → Ⅎyψ)
8	2, 4, 5, 7	sbciedf 3082	. . 3 ⊢ ((B ∈ W ∧ x = A) → ([̣B / y]̣φ ↔ ψ))
9	8	adantll 694	. 2 ⊢ (((A ∈ V ∧ B ∈ W) ∧ x = A) → ([̣B / y]̣φ ↔ ψ))
10		nfv 1619	. . 3 ⊢ Ⅎx A ∈ V
11		sbc2iegf.3	. . 3 ⊢ Ⅎx B ∈ W
12	10, 11	nfan 1824	. 2 ⊢ Ⅎx(A ∈ V ∧ B ∈ W)
13		sbc2iegf.1	. . 3 ⊢ Ⅎxψ
14	13	a1i 10	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → Ⅎxψ)
15	1, 9, 12, 14	sbciedf 3082	1 ⊢ ((A ∈ V ∧ B ∈ W) → ([̣A / x]̣[̣B / y]̣φ ↔ ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is used by:  sbc2ie  3114  opelopabaf  4711