Theorem sbcrext 3119

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbcrext	⊢ ((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 sbcrext

Step	Hyp	Ref	Expression
1		elex 2867	. 2 ⊢ (A ∈ V → A ∈ V)
2		sbcng 3086	. . . . 5 ⊢ (A ∈ V → ([̣A / x]̣ ¬ ∀y ∈ B ¬ φ ↔ ¬ [̣A / x]̣∀y ∈ B ¬ φ))
3	2	adantr 451	. . . 4 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣ ¬ ∀y ∈ B ¬ φ ↔ ¬ [̣A / x]̣∀y ∈ B ¬ φ))
4		sbcralt 3118	. . . . . 6 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣∀y ∈ B ¬ φ ↔ ∀y ∈ B [̣A / x]̣ ¬ φ))
5		nfnfc1 2492	. . . . . . . . 9 ⊢ ℲyℲyA
6		id 19	. . . . . . . . . 10 ⊢ (ℲyA → ℲyA)
7		nfcvd 2490	. . . . . . . . . 10 ⊢ (ℲyA → ℲyV)
8	6, 7	nfeld 2504	. . . . . . . . 9 ⊢ (ℲyA → Ⅎy A ∈ V)
9	5, 8	nfan1 1881	. . . . . . . 8 ⊢ Ⅎy(ℲyA ∧ A ∈ V)
10		sbcng 3086	. . . . . . . . 9 ⊢ (A ∈ V → ([̣A / x]̣ ¬ φ ↔ ¬ [̣A / x]̣φ))
11	10	adantl 452	. . . . . . . 8 ⊢ ((ℲyA ∧ A ∈ V) → ([̣A / x]̣ ¬ φ ↔ ¬ [̣A / x]̣φ))
12	9, 11	ralbid 2632	. . . . . . 7 ⊢ ((ℲyA ∧ A ∈ V) → (∀y ∈ B [̣A / x]̣ ¬ φ ↔ ∀y ∈ B ¬ [̣A / x]̣φ))
13	12	ancoms 439	. . . . . 6 ⊢ ((A ∈ V ∧ ℲyA) → (∀y ∈ B [̣A / x]̣ ¬ φ ↔ ∀y ∈ B ¬ [̣A / x]̣φ))
14	4, 13	bitrd 244	. . . . 5 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣∀y ∈ B ¬ φ ↔ ∀y ∈ B ¬ [̣A / x]̣φ))
15	14	notbid 285	. . . 4 ⊢ ((A ∈ V ∧ ℲyA) → (¬ [̣A / x]̣∀y ∈ B ¬ φ ↔ ¬ ∀y ∈ B ¬ [̣A / x]̣φ))
16	3, 15	bitrd 244	. . 3 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣ ¬ ∀y ∈ B ¬ φ ↔ ¬ ∀y ∈ B ¬ [̣A / x]̣φ))
17		dfrex2 2627	. . . 4 ⊢ (∃y ∈ B φ ↔ ¬ ∀y ∈ B ¬ φ)
18	17	sbcbii 3101	. . 3 ⊢ ([̣A / x]̣∃y ∈ B φ ↔ [̣A / x]̣ ¬ ∀y ∈ B ¬ φ)
19		dfrex2 2627	. . 3 ⊢ (∃y ∈ B [̣A / x]̣φ ↔ ¬ ∀y ∈ B ¬ [̣A / x]̣φ)
20	16, 18, 19	3bitr4g 279	. 2 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣∃y ∈ B φ ↔ ∃y ∈ B [̣A / x]̣φ))
21	1, 20	sylan 457	1 ⊢ ((A ∈ V ∧ ℲyA) → ([̣A / x]̣∃y ∈ B φ ↔ ∃y ∈ B [̣A / x]̣φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614  ∃wrex 2615  Vcvv 2859  [̣wsbc 3046

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 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047

This theorem is referenced by: (None)