Theorem sbcrextOLD 3479

Description: Obsolete proof of sbcrext 3478 as of 7-Jul-2021. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcrextOLD	⊢ (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)

Proof of Theorem sbcrextOLD

Step	Hyp	Ref	Expression
1		sbcng 3443	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
3		sbcralt 3477	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑))
4		nfnfc1 2754	. . . . . . . . 9 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
5		id 22	. . . . . . . . . 10 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
6		nfcvd 2752	. . . . . . . . . 10 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦V)
7	5, 6	nfeld 2759	. . . . . . . . 9 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
8	4, 7	nfan1 2056	. . . . . . . 8 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V)
9		sbcng 3443	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
10	9	adantl 481	. . . . . . . 8 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
11	8, 10	ralbid 2966	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
12	11	ancoms 468	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
13	3, 12	bitrd 267	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
14	13	notbid 307	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → (¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
15	2, 14	bitrd 267	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
16		dfrex2 2979	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
17	16	sbcbii 3458	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
18		dfrex2 2979	. . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)
19	15, 17, 18	3bitr4g 302	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
20		sbcex 3412	. . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
21	20	con3i 149	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
22	21	adantr 480	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ¬ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
23		sbcex 3412	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
24	23	2a1i 12	. . . . . 6 ⊢ (Ⅎ𝑦𝐴 → (𝑦 ∈ 𝐵 → ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)))
25	4, 7, 24	rexlimd2 3007	. . . . 5 ⊢ (Ⅎ𝑦𝐴 → (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V))
26	25	con3rr3 150	. . . 4 ⊢ (¬ 𝐴 ∈ V → (Ⅎ𝑦𝐴 → ¬ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
27	26	imp 444	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ¬ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
28	22, 27	2falsed 365	. 2 ⊢ ((¬ 𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
29	19, 28	pm2.61ian 827	1 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)