Theorem bj-mo3OLD 32022

Description: Obsolete proof temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-mo3OLD.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
bj-mo3OLD	⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-mo3OLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2465	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		bj-mo3OLD.nf	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
3		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 = 𝑧
4	2, 3	nfim 1813	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧)
5		nfs1v 2425	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
6		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 = 𝑧
7	5, 6	nfim 1813	. . . . . . . 8 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)
8		sbequ2 1869	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
9		ax7 1930	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
10	8, 9	imim12d 79	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
11	4, 7, 10	cbv3 2253	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧))
12	11	ancli 572	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∀𝑥(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
13	4, 7	aaan 2156	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) ↔ (∀𝑥(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
14	12, 13	sylibr 223	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
15		prth 593	. . . . . . 7 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
16		equtr2 1941	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
17	15, 16	syl6 34	. . . . . 6 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
18	17	2alimi 1731	. . . . 5 ⊢ (∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
19	14, 18	syl 17	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
20	19	exlimiv 1845	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
21	1, 20	sylbi 206	. 2 ⊢ (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
22		nfa1 2015	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)
23		pm3.3 459	. . . . . . . . . 10 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
24	23	com3r 85	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝜑 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)))
25	5, 24	alimd 2068	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
26	25	com12 32	. . . . . . 7 ⊢ (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
27	26	sps 2043	. . . . . 6 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
28	22, 27	eximd 2072	. . . . 5 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
29	2	sb8e 2413	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
30	2	mo2 2467	. . . . 5 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
31	28, 29, 30	3imtr4g 284	. . . 4 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃*𝑥𝜑))
32		moabs 2489	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
33	31, 32	sylibr 223	. . 3 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
34	33	alcoms 2022	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
35	21, 34	impbii 198	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  [wsb 1867  ∃*wmo 2459

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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463

This theorem is referenced by: (None)