Theorem sbc2ie 3114

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
sbc2ie.1	⊢ A ∈ V
sbc2ie.2	⊢ B ∈ V
sbc2ie.3	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))

Assertion

Ref	Expression
sbc2ie	⊢ ([̣A / x]̣[̣B / y]̣φ ↔ ψ)

Distinct variable groups:   x,y,A   y,B   ψ,x,y

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

Proof of Theorem sbc2ie

Step	Hyp	Ref	Expression
1		sbc2ie.1	. 2 ⊢ A ∈ V
2		sbc2ie.2	. 2 ⊢ B ∈ V
3		nfv 1619	. . 3 ⊢ Ⅎxψ
4		nfv 1619	. . 3 ⊢ Ⅎyψ
5	2	nfth 1553	. . 3 ⊢ Ⅎx B ∈ V
6		sbc2ie.3	. . 3 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
7	3, 4, 5, 6	sbc2iegf 3113	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣A / x]̣[̣B / y]̣φ ↔ ψ))
8	1, 2, 7	mp2an 653	1 ⊢ ([̣A / x]̣[̣B / y]̣φ ↔ ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [̣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:  sbc3ie  3116