Theorem mo5f 28708

Description: Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)

Hypotheses

Ref	Expression
mo5f.1	⊢ Ⅎ𝑖𝜑
mo5f.2	⊢ Ⅎ𝑗𝜑

Assertion

Ref	Expression
mo5f	⊢ (∃*𝑥𝜑 ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))

Distinct variable group:   𝑖,𝑗,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑖,𝑗)

Proof of Theorem mo5f

Step	Hyp	Ref	Expression
1		mo5f.2	. . 3 ⊢ Ⅎ𝑗𝜑
2	1	mo3 2495	. 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
3		mo5f.1	. . . . . 6 ⊢ Ⅎ𝑖𝜑
4	3	nfsb 2428	. . . . . 6 ⊢ Ⅎ𝑖[𝑗 / 𝑥]𝜑
5	3, 4	nfan 1816	. . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ [𝑗 / 𝑥]𝜑)
6		nfv 1830	. . . . 5 ⊢ Ⅎ𝑖 𝑥 = 𝑗
7	5, 6	nfim 1813	. . . 4 ⊢ Ⅎ𝑖((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
8	7	nfal 2139	. . 3 ⊢ Ⅎ𝑖∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
9	8	sb8 2412	. 2 ⊢ (∀𝑥∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
10		sbim 2383	. . . . 5 ⊢ ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗))
11		sban 2387	. . . . . . 7 ⊢ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
12		nfs1v 2425	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑗 / 𝑥]𝜑
13	12	sbf 2368	. . . . . . . . 9 ⊢ ([𝑖 / 𝑥][𝑗 / 𝑥]𝜑 ↔ [𝑗 / 𝑥]𝜑)
14	13	bicomi 213	. . . . . . . 8 ⊢ ([𝑗 / 𝑥]𝜑 ↔ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑)
15	14	anbi2i 726	. . . . . . 7 ⊢ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
16	11, 15	bitr4i 266	. . . . . 6 ⊢ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑))
17		equsb3 2420	. . . . . 6 ⊢ ([𝑖 / 𝑥]𝑥 = 𝑗 ↔ 𝑖 = 𝑗)
18	16, 17	imbi12i 339	. . . . 5 ⊢ (([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
19	10, 18	bitri 263	. . . 4 ⊢ ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
20	19	sbalv 2452	. . 3 ⊢ ([𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
21	20	albii 1737	. 2 ⊢ (∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
22	2, 9, 21	3bitri 285	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  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)