Theorem sbc2iedv 3115

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 10	. 2 ⊢ (φ → A ∈ V)
3		sbc2iedv.2	. . . 4 ⊢ B ∈ V
4	3	a1i 10	. . 3 ⊢ ((φ ∧ x = A) → B ∈ V)
5		sbc2iedv.3	. . . 4 ⊢ (φ → ((x = A ∧ y = B) → (ψ ↔ χ)))
6	5	impl 603	. . 3 ⊢ (((φ ∧ x = A) ∧ y = B) → (ψ ↔ χ))
7	4, 6	sbcied 3083	. 2 ⊢ ((φ ∧ x = A) → ([̣B / y]̣ψ ↔ χ))
8	2, 7	sbcied 3083	1 ⊢ (φ → ([̣A / x]̣[̣B / y]̣ψ ↔ χ))

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: (None)