Theorem sbcralt 3510

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 3458	. 2 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
2		simpl 473	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → 𝐴 ∈ 𝑉)
3		sbsbc 3439	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
4		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
5		nfs1v 2437	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfral 2945	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 2111	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbie 2408	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
10	3, 9	bitr3i 266	. . . 4 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
11		nfnfc1 2767	. . . . . . 7 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
12		nfcvd 2765	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝑧)
13		id 22	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
14	12, 13	nfeqd 2772	. . . . . . 7 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
15	11, 14	nfan1 2068	. . . . . 6 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴)
16		dfsbcq2 3438	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
17	16	adantl 482	. . . . . 6 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
18	15, 17	ralbid 2983	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
19	18	adantll 750	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
20	10, 19	syl5bb 272	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
21	2, 20	sbcied 3472	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
22	1, 21	syl5bbr 274	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  [wsb 1880   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcrext  3511  sbcrextOLD  3512  sbcralg  3513