Theorem sbcralt 3801

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 3743	. 2 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
2		simpl 486	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → 𝐴 ∈ 𝑉)
3		sbsbc 3724	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
4		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
5		nfs1v 2157	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfralw 3189	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 2250	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	ralbidv 3162	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbiev 2322	. . . . 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 2958	. . . . . . 7 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
12		nfcvd 2956	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝑧)
13		id 22	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
14	12, 13	nfeqd 2965	. . . . . . 7 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
15	11, 14	nfan1 2198	. . . . . 6 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴)
16		dfsbcq2 3723	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
17	16	adantl 485	. . . . . 6 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
18	15, 17	ralbid 3195	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
19	18	adantll 713	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
20	10, 19	syl5bb 286	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
21	2, 20	sbcied 3762	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  [wsb 2069   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-sbc 3721

This theorem is referenced by:  sbcrext  3802  sbcralg  3803