Theorem mo5f 31729

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 2559	. 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
3		mo5f.1	. . . . . 6 ⊢ Ⅎ𝑖𝜑
4	3	nfsbv 2324	. . . . . 6 ⊢ Ⅎ𝑖[𝑗 / 𝑥]𝜑
5	3, 4	nfan 1903	. . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ [𝑗 / 𝑥]𝜑)
6		nfv 1918	. . . . 5 ⊢ Ⅎ𝑖 𝑥 = 𝑗
7	5, 6	nfim 1900	. . . 4 ⊢ Ⅎ𝑖((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
8	7	nfal 2317	. . 3 ⊢ Ⅎ𝑖∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
9	8	sb8f 2350	. 2 ⊢ (∀𝑥∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
10		sbim 2300	. . . . 5 ⊢ ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗))
11		sban 2084	. . . . . . 7 ⊢ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
12		nfs1v 2154	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑗 / 𝑥]𝜑
13	12	sbf 2263	. . . . . . . . 9 ⊢ ([𝑖 / 𝑥][𝑗 / 𝑥]𝜑 ↔ [𝑗 / 𝑥]𝜑)
14	13	bicomi 223	. . . . . . . 8 ⊢ ([𝑗 / 𝑥]𝜑 ↔ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑)
15	14	anbi2i 624	. . . . . . 7 ⊢ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
16	11, 15	bitr4i 278	. . . . . 6 ⊢ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑))
17		equsb3 2102	. . . . . 6 ⊢ ([𝑖 / 𝑥]𝑥 = 𝑗 ↔ 𝑖 = 𝑗)
18	16, 17	imbi12i 351	. . . . 5 ⊢ (([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
19	10, 18	bitri 275	. . . 4 ⊢ ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
20	19	sbalv 2161	. . 3 ⊢ ([𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
21	20	albii 1822	. 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 297	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540  Ⅎwnf 1786  [wsb 2068  ∃*wmo 2533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535

This theorem is referenced by: (None)