Theorem sbcrextOLD 3512

Description: Obsolete proof of sbcrext 3511 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 3476	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
2	1	adantr 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
3		sbcralt 3510	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑))
4		nfnfc1 2767	. . . . . . . . 9 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
5		id 22	. . . . . . . . . 10 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
6		nfcvd 2765	. . . . . . . . . 10 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦V)
7	5, 6	nfeld 2773	. . . . . . . . 9 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
8	4, 7	nfan1 2068	. . . . . . . 8 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V)
9		sbcng 3476	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
10	9	adantl 482	. . . . . . . 8 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
11	8, 10	ralbid 2983	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
12	11	ancoms 469	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
13	3, 12	bitrd 268	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
14	13	notbid 308	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → (¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
15	2, 14	bitrd 268	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
16		dfrex2 2996	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
17	16	sbcbii 3491	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
18		dfrex2 2996	. . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)
19	15, 17, 18	3bitr4g 303	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
20		sbcex 3445	. . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
21	20	con3i 150	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
22	21	adantr 481	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ¬ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
23		sbcex 3445	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
24	23	2a1i 12	. . . . . 6 ⊢ (Ⅎ𝑦𝐴 → (𝑦 ∈ 𝐵 → ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)))
25	4, 7, 24	rexlimd2 3025	. . . . 5 ⊢ (Ⅎ𝑦𝐴 → (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V))
26	25	con3rr3 151	. . . 4 ⊢ (¬ 𝐴 ∈ V → (Ⅎ𝑦𝐴 → ¬ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
27	26	imp 445	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ¬ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
28	22, 27	2falsed 366	. 2 ⊢ ((¬ 𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
29	19, 28	pm2.61ian 831	1 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  ∃wrex 2913  Vcvv 3200  [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-rex 2918  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)