Theorem sbcralt 3119

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Assertion

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

Distinct variable groups:   x,y   x,B

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

Proof of Theorem sbcralt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcco 3069	. 2 ⊢ ([̣A / z]̣[̣z / x]̣∀y ∈ B φ ↔ [̣A / x]̣∀y ∈ B φ)
2		simpl 443	. . 3 ⊢ ((A ∈ V ∧ ℲyA) → A ∈ V)
3		sbsbc 3051	. . . . 5 ⊢ ([z / x]∀y ∈ B φ ↔ [̣z / x]̣∀y ∈ B φ)
4		nfcv 2490	. . . . . . 7 ⊢ ℲxB
5		nfs1v 2106	. . . . . . 7 ⊢ Ⅎx[z / x]φ
6	4, 5	nfral 2668	. . . . . 6 ⊢ Ⅎx∀y ∈ B [z / x]φ
7		sbequ12 1919	. . . . . . 7 ⊢ (x = z → (φ ↔ [z / x]φ))
8	7	ralbidv 2635	. . . . . 6 ⊢ (x = z → (∀y ∈ B φ ↔ ∀y ∈ B [z / x]φ))
9	6, 8	sbie 2038	. . . . 5 ⊢ ([z / x]∀y ∈ B φ ↔ ∀y ∈ B [z / x]φ)
10	3, 9	bitr3i 242	. . . 4 ⊢ ([̣z / x]̣∀y ∈ B φ ↔ ∀y ∈ B [z / x]φ)
11		nfnfc1 2493	. . . . . . 7 ⊢ ℲyℲyA
12		nfcvd 2491	. . . . . . . 8 ⊢ (ℲyA → Ⅎyz)
13		id 19	. . . . . . . 8 ⊢ (ℲyA → ℲyA)
14	12, 13	nfeqd 2504	. . . . . . 7 ⊢ (ℲyA → Ⅎy z = A)
15	11, 14	nfan1 1881	. . . . . 6 ⊢ Ⅎy(ℲyA ∧ z = A)
16		dfsbcq2 3050	. . . . . . 7 ⊢ (z = A → ([z / x]φ ↔ [̣A / x]̣φ))
17	16	adantl 452	. . . . . 6 ⊢ ((ℲyA ∧ z = A) → ([z / x]φ ↔ [̣A / x]̣φ))
18	15, 17	ralbid 2633	. . . . 5 ⊢ ((ℲyA ∧ z = A) → (∀y ∈ B [z / x]φ ↔ ∀y ∈ B [̣A / x]̣φ))
19	18	adantll 694	. . . 4 ⊢ (((A ∈ V ∧ ℲyA) ∧ z = A) → (∀y ∈ B [z / x]φ ↔ ∀y ∈ B [̣A / x]̣φ))
20	10, 19	syl5bb 248	. . 3 ⊢ (((A ∈ V ∧ ℲyA) ∧ z = A) → ([̣z / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ))
21	2, 20	sbcied 3083	. 2 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / z]̣[̣z / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ))
22	1, 21	syl5bbr 250	1 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2477  ∀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-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-ral 2620  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbcrext  3120