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Theorem reuprg 4631
Description: Convert a restricted existential uniqueness over a pair to a disjunction and an implication . (Contributed by AV, 2-Apr-2023.)
Hypotheses
Ref Expression
reuprg.1 (𝑥 = 𝐴 → (𝜑𝜓))
reuprg.2 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
reuprg ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem reuprg
StepHypRef Expression
1 reuprg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
2 reuprg.2 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
31, 2reuprg0 4630 . 2 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵)))))
4 orddi 1003 . . 3 (((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵))) ↔ (((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ∧ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))))
5 curryax 887 . . . . . 6 (𝜓 ∨ (𝜓𝐴 = 𝐵))
65biantru 530 . . . . 5 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))))
76bicomi 225 . . . 4 (((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ↔ (𝜓𝜒))
8 curryax 887 . . . . . . . 8 (𝜒 ∨ (𝜒𝐴 = 𝐵))
9 orcom 864 . . . . . . . 8 (((𝜒𝐴 = 𝐵) ∨ 𝜒) ↔ (𝜒 ∨ (𝜒𝐴 = 𝐵)))
108, 9mpbir 232 . . . . . . 7 ((𝜒𝐴 = 𝐵) ∨ 𝜒)
1110biantrur 531 . . . . . 6 (((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)) ↔ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))))
1211bicomi 225 . . . . 5 ((((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))) ↔ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))
13 pm4.79 997 . . . . 5 (((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)) ↔ ((𝜒𝜓) → 𝐴 = 𝐵))
1412, 13bitri 276 . . . 4 ((((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))) ↔ ((𝜒𝜓) → 𝐴 = 𝐵))
157, 14anbi12i 626 . . 3 ((((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ∧ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))) ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)))
164, 15bitri 276 . 2 (((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵))) ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)))
173, 16syl6bb 288 1 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  ∃!wreu 3137  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-reu 3142  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  reurexprg  4632
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