Theorem sbcralt 3823

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		sbccow 3765	. 2 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
2		simpl 488	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → 𝐴 ∈ 𝑉)
3		sbsbc 3746	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
4		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
5		nfs1v 2196	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfralw 3313	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 2290	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	ralbidv 3189	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbiev 2349	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
10	3, 9	bitr3i 280	. . . 4 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
11		nfnfc1 2929	. . . . . . 7 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
12		nfcvd 2927	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝑧)
13		id 23	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
14	12, 13	nfeqd 2936	. . . . . . 7 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
15	11, 14	nfan1 2240	. . . . . 6 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴)
16		dfsbcq2 3745	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
17	16	adantl 487	. . . . . 6 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
18	15, 17	ralbid 3279	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
19	18	adantll 727	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
20	10, 19	bitrid 286	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
21	2, 20	sbcied 3785	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
22	1, 21	bitr3id 288	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  [wsb 2101   ∈ wcel 2147  Ⅎwnfc 2911  ∀wral 3080  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-v 3456  df-sbc 3743

This theorem is used by:  sbcrext  3824  sbcralg  3825  modelaxreplem3  46234