Theorem sbc3ie 3116

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbc3ie.1	⊢ A ∈ V
sbc3ie.2	⊢ B ∈ V
sbc3ie.3	⊢ C ∈ V
sbc3ie.4	⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))

Assertion

Ref	Expression
sbc3ie	⊢ ([̣A / x]̣[̣B / y]̣[̣C / z]̣φ ↔ ψ)

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

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

Proof of Theorem sbc3ie

Step	Hyp	Ref	Expression
1		sbc3ie.1	. 2 ⊢ A ∈ V
2		sbc3ie.2	. 2 ⊢ B ∈ V
3		sbc3ie.3	. . . 4 ⊢ C ∈ V
4	3	a1i 10	. . 3 ⊢ ((x = A ∧ y = B) → C ∈ V)
5		sbc3ie.4	. . . 4 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
6	5	3expa 1151	. . 3 ⊢ (((x = A ∧ y = B) ∧ z = C) → (φ ↔ ψ))
7	4, 6	sbcied 3083	. 2 ⊢ ((x = A ∧ y = B) → ([̣C / z]̣φ ↔ ψ))
8	1, 2, 7	sbc2ie 3114	1 ⊢ ([̣A / x]̣[̣B / y]̣[̣C / z]̣φ ↔ ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = 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)