Theorem sbcralg 3121

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcralg	⊢ (A ∈ V → ([̣A / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ))

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

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

Proof of Theorem sbcralg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. 2 ⊢ (z = A → ([z / x]∀y ∈ B φ ↔ [̣A / x]̣∀y ∈ B φ))
2		dfsbcq2 3050	. . 3 ⊢ (z = A → ([z / x]φ ↔ [̣A / x]̣φ))
3	2	ralbidv 2635	. 2 ⊢ (z = A → (∀y ∈ B [z / x]φ ↔ ∀y ∈ B [̣A / x]̣φ))
4		nfcv 2490	. . . 4 ⊢ ℲxB
5		nfs1v 2106	. . . 4 ⊢ Ⅎx[z / x]φ
6	4, 5	nfral 2668	. . 3 ⊢ Ⅎx∀y ∈ B [z / x]φ
7		sbequ12 1919	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
8	7	ralbidv 2635	. . 3 ⊢ (x = z → (∀y ∈ B φ ↔ ∀y ∈ B [z / x]φ))
9	6, 8	sbie 2038	. 2 ⊢ ([z / x]∀y ∈ B φ ↔ ∀y ∈ B [z / x]φ)
10	1, 3, 9	vtoclbg 2916	1 ⊢ (A ∈ V → ([̣A / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∀wral 2615  [̣wsbc 3047

This theorem was proved from 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 theorem 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-ral 2620  df-v 2862  df-sbc 3048

This theorem is referenced by:  r19.12sn  3790