Theorem sbcie2g 3080

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3081 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
sbcie2g.1	⊢ (x = y → (φ ↔ ψ))
sbcie2g.2	⊢ (y = A → (ψ ↔ χ))

Assertion

Ref	Expression
sbcie2g	⊢ (A ∈ V → ([̣A / x]̣φ ↔ χ))

Distinct variable groups:   x,y   y,A   χ,y   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)   χ(x)   A(x)   V(x, y)

Proof of Theorem sbcie2g

Step	Hyp	Ref	Expression
1		dfsbcq 3049	. 2 ⊢ (y = A → ([̣y / x]̣φ ↔ [̣A / x]̣φ))
2		sbcie2g.2	. 2 ⊢ (y = A → (ψ ↔ χ))
3		sbsbc 3051	. . 3 ⊢ ([y / x]φ ↔ [̣y / x]̣φ)
4		nfv 1619	. . . 4 ⊢ Ⅎxψ
5		sbcie2g.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
6	4, 5	sbie 2038	. . 3 ⊢ ([y / x]φ ↔ ψ)
7	3, 6	bitr3i 242	. 2 ⊢ ([̣y / x]̣φ ↔ ψ)
8	1, 2, 7	vtoclbg 2916	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ 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-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:  csbie2g  3183  brab1  4685