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Theorem rmo4f 3563
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 2009 . . 3 𝑦𝜑
41, 2, 3rmo3f 3562 . 2 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5 rmo4f.3 . . . . . 6 𝑥𝜓
6 rmo4f.4 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
75, 6sbie 2499 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
87anbi2i 616 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
98imbi1i 340 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
1092ralbii 3128 . 2 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
114, 10bitri 266 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wnf 1878  [wsb 2061  wnfc 2894  wral 3055  ∃*wrmo 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-clel 2761  df-nfc 2896  df-ral 3060  df-rmo 3063
This theorem is referenced by:  disjorf  29775  funcnv5mpt  29853
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