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Theorem rmo3 3818
 Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) Avoid ax-13 2379. (Revised by Wolf Lammen, 30-Apr-2023.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo3 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo3
StepHypRef Expression
1 df-rmo 3114 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 sban 2085 . . . . . . . . . . 11 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑦 / 𝑥]𝑥𝐴 ∧ [𝑦 / 𝑥]𝜑))
3 clelsb3 2917 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
43anbi1i 626 . . . . . . . . . . 11 (([𝑦 / 𝑥]𝑥𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
52, 4bitri 278 . . . . . . . . . 10 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
65anbi2i 625 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
7 an4 655 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
8 ancom 464 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
98anbi1i 626 . . . . . . . . 9 (((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
106, 7, 93bitri 300 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1110imbi1i 353 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
12 impexp 454 . . . . . . 7 ((((𝑦𝐴𝑥𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑦𝐴𝑥𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
13 impexp 454 . . . . . . 7 (((𝑦𝐴𝑥𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
1411, 12, 133bitri 300 . . . . . 6 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
1514albii 1821 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
16 df-ral 3111 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
17 r19.21v 3142 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
1815, 16, 173bitr2i 302 . . . 4 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
1918albii 1821 . . 3 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
20 nfv 1915 . . . . 5 𝑦 𝑥𝐴
21 rmo2.1 . . . . 5 𝑦𝜑
2220, 21nfan 1900 . . . 4 𝑦(𝑥𝐴𝜑)
2322mo3 2623 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦))
24 df-ral 3111 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2519, 23, 243bitr4i 306 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
261, 25bitri 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  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  ∃*wmo 2596  ∀wral 3106  ∃*wrmo 3109 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175 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 2070  df-mo 2598  df-clel 2870  df-ral 3111  df-rmo 3114 This theorem is referenced by:  poimirlem25  35101  poimirlem26  35102
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