Theorem sbcralt 3819

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 3760	. 2 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
2		simpl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → 𝐴 ∈ 𝑉)
3		sbsbc 3741	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
4		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
5		nfs1v 2161	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfralw 3280	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 2256	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	ralbidv 3156	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbiev 2317	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
10	3, 9	bitr3i 277	. . . 4 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
11		nfnfc1 2898	. . . . . . 7 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
12		nfcvd 2896	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝑧)
13		id 22	. . . . . . . 8 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
14	12, 13	nfeqd 2906	. . . . . . 7 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
15	11, 14	nfan1 2205	. . . . . 6 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴)
16		dfsbcq2 3740	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
17	16	adantl 481	. . . . . 6 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
18	15, 17	ralbid 3246	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
19	18	adantll 714	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
20	10, 19	bitrid 283	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
21	2, 20	sbcied 3781	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
22	1, 21	bitr3id 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  [wsb 2067   ∈ wcel 2113  Ⅎwnfc 2880  ∀wral 3048  [wsbc 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-v 3439  df-sbc 3738

This theorem is referenced by:  sbcrext  3820  sbcralg  3821  modelaxreplem3  45100