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Theorem rmo3f 3711
Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmo3f.1 𝑥𝐴
rmo3f.2 𝑦𝐴
rmo3f.3 𝑦𝜑
Assertion
Ref Expression
rmo3f (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem rmo3f
StepHypRef Expression
1 df-rmo 3141 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 sban 2087 . . . . . . . . . . 11 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑦 / 𝑥]𝑥𝐴 ∧ [𝑦 / 𝑥]𝜑))
3 rmo3f.1 . . . . . . . . . . . . 13 𝑥𝐴
43clelsb3fw 2986 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
54anbi1i 626 . . . . . . . . . . 11 (([𝑦 / 𝑥]𝑥𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
62, 5bitri 278 . . . . . . . . . 10 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
76anbi2i 625 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
8 an4 655 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
9 ancom 464 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
109anbi1i 626 . . . . . . . . 9 (((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
117, 8, 103bitri 300 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1211imbi1i 353 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
13 impexp 454 . . . . . . 7 ((((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑦𝐴𝑥𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
14 impexp 454 . . . . . . 7 (((𝑦𝐴𝑥𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
1512, 13, 143bitri 300 . . . . . 6 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
1615albii 1821 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
17 df-ral 3138 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
18 rmo3f.2 . . . . . . 7 𝑦𝐴
1918nfcri 2969 . . . . . 6 𝑦 𝑥𝐴
2019r19.21 3209 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2116, 17, 203bitr2i 302 . . . 4 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2221albii 1821 . . 3 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
23 rmo3f.3 . . . . 5 𝑦𝜑
2419, 23nfan 1901 . . . 4 𝑦(𝑥𝐴𝜑)
2524mo3 2649 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦))
26 df-ral 3138 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2722, 25, 263bitr4i 306 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
281, 27bitri 278 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wnf 1785  [wsb 2070  wcel 2115  ∃*wmo 2622  wnfc 2962  wral 3133  ∃*wrmo 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-clel 2896  df-nfc 2964  df-ral 3138  df-rmo 3141
This theorem is referenced by:  rmo4f  3712
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