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Theorem rmo4f 2883
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 1509 . . 3 𝑦𝜑
41, 2, 3rmo3f 2882 . 2 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5 rmo4f.3 . . . . . 6 𝑥𝜓
6 rmo4f.4 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
75, 6sbie 1765 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
87anbi2i 453 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
98imbi1i 237 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
1092ralbii 2444 . 2 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
114, 10bitri 183 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wnf 1437  [wsb 1736  wnfc 2269  wral 2417  ∃*wrmo 2420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rmo 2425
This theorem is referenced by:  disjxp1  6137
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