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Theorem mo5f 29646
Description: Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
Hypotheses
Ref Expression
mo5f.1 𝑖𝜑
mo5f.2 𝑗𝜑
Assertion
Ref Expression
mo5f (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([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
StepHypRef Expression
1 mo5f.2 . . 3 𝑗𝜑
21mo3 2667 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
3 mo5f.1 . . . . . 6 𝑖𝜑
43nfsb 2601 . . . . . 6 𝑖[𝑗 / 𝑥]𝜑
53, 4nfan 1990 . . . . 5 𝑖(𝜑 ∧ [𝑗 / 𝑥]𝜑)
6 nfv 2005 . . . . 5 𝑖 𝑥 = 𝑗
75, 6nfim 1987 . . . 4 𝑖((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
87nfal 2328 . . 3 𝑖𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
98sb8 2582 . 2 (∀𝑥𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
10 sbim 2553 . . . . 5 ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗))
11 sban 2557 . . . . . . 7 ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
12 nfs1v 2285 . . . . . . . . . 10 𝑥[𝑗 / 𝑥]𝜑
1312sbf 2538 . . . . . . . . 9 ([𝑖 / 𝑥][𝑗 / 𝑥]𝜑 ↔ [𝑗 / 𝑥]𝜑)
1413bicomi 215 . . . . . . . 8 ([𝑗 / 𝑥]𝜑 ↔ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑)
1514anbi2i 611 . . . . . . 7 (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
1611, 15bitr4i 269 . . . . . 6 ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑))
17 equsb3 2591 . . . . . 6 ([𝑖 / 𝑥]𝑥 = 𝑗𝑖 = 𝑗)
1816, 17imbi12i 341 . . . . 5 (([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
1910, 18bitri 266 . . . 4 ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
2019sbalv 2623 . . 3 ([𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
2120albii 1904 . 2 (∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
222, 9, 213bitri 288 1 (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635  wnf 1863  [wsb 2059  ∃*wmo 2630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634
This theorem is referenced by: (None)
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