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Theorem mo5f 29664
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 2656 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
3 mo5f.1 . . . . . 6 𝑖𝜑
43nfsb 2590 . . . . . 6 𝑖[𝑗 / 𝑥]𝜑
53, 4nfan 1980 . . . . 5 𝑖(𝜑 ∧ [𝑗 / 𝑥]𝜑)
6 nfv 1995 . . . . 5 𝑖 𝑥 = 𝑗
75, 6nfim 1977 . . . 4 𝑖((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
87nfal 2317 . . 3 𝑖𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗)
98sb8 2571 . 2 (∀𝑥𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗))
10 sbim 2542 . . . . 5 ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗))
11 sban 2546 . . . . . . 7 ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
12 nfs1v 2274 . . . . . . . . . 10 𝑥[𝑗 / 𝑥]𝜑
1312sbf 2527 . . . . . . . . 9 ([𝑖 / 𝑥][𝑗 / 𝑥]𝜑 ↔ [𝑗 / 𝑥]𝜑)
1413bicomi 214 . . . . . . . 8 ([𝑗 / 𝑥]𝜑 ↔ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑)
1514anbi2i 609 . . . . . . 7 (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥][𝑗 / 𝑥]𝜑))
1611, 15bitr4i 267 . . . . . 6 ([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑))
17 equsb3 2580 . . . . . 6 ([𝑖 / 𝑥]𝑥 = 𝑗𝑖 = 𝑗)
1816, 17imbi12i 339 . . . . 5 (([𝑖 / 𝑥](𝜑 ∧ [𝑗 / 𝑥]𝜑) → [𝑖 / 𝑥]𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
1910, 18bitri 264 . . . 4 ([𝑖 / 𝑥]((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ (([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
2019sbalv 2612 . . 3 ([𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
2120albii 1895 . 2 (∀𝑖[𝑖 / 𝑥]∀𝑗((𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑥 = 𝑗) ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
222, 9, 213bitri 286 1 (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wnf 1856  [wsb 2049  ∃*wmo 2619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623
This theorem is referenced by: (None)
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