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Theorem rmo4f 2954
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmo4f.1 𝑥𝐴
rmo4f.2 𝑦𝐴
rmo4f.3 𝑥𝜓
rmo4f.4 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
rmo4f (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem rmo4f
StepHypRef Expression
1 rmo4f.1 . . 3 𝑥𝐴
2 rmo4f.2 . . 3 𝑦𝐴
3 nfv 1539 . . 3 𝑦𝜑
41, 2, 3rmo3f 2953 . 2 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5 rmo4f.3 . . . . . 6 𝑥𝜓
6 rmo4f.4 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
75, 6sbie 1802 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
87anbi2i 457 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
98imbi1i 238 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
1092ralbii 2498 . 2 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
114, 10bitri 184 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wnf 1471  [wsb 1773  wnfc 2319  wral 2468  ∃*wrmo 2471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rmo 2476
This theorem is referenced by:  disjxp1  6276
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