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Theorem reuprg 4624
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 4623 . 2 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵)))))
4 orddi 1007 . . 3 (((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵))) ↔ (((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ∧ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))))
5 curryax 891 . . . . . 6 (𝜓 ∨ (𝜓𝐴 = 𝐵))
65biantru 533 . . . . 5 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))))
76bicomi 227 . . . 4 (((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ↔ (𝜓𝜒))
8 curryax 891 . . . . . . . 8 (𝜒 ∨ (𝜒𝐴 = 𝐵))
9 orcom 867 . . . . . . . 8 (((𝜒𝐴 = 𝐵) ∨ 𝜒) ↔ (𝜒 ∨ (𝜒𝐴 = 𝐵)))
108, 9mpbir 234 . . . . . . 7 ((𝜒𝐴 = 𝐵) ∨ 𝜒)
1110biantrur 534 . . . . . 6 (((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)) ↔ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))))
1211bicomi 227 . . . . 5 ((((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))) ↔ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))
13 pm4.79 1001 . . . . 5 (((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)) ↔ ((𝜒𝜓) → 𝐴 = 𝐵))
1412, 13bitri 278 . . . 4 ((((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))) ↔ ((𝜒𝜓) → 𝐴 = 𝐵))
157, 14anbi12i 629 . . 3 ((((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ∧ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))) ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)))
164, 15bitri 278 . 2 (((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵))) ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)))
173, 16syl6bb 290 1 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  ∃!wreu 3135  {cpr 4552
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-reu 3140  df-v 3482  df-sbc 3759  df-un 3924  df-sn 4551  df-pr 4553
This theorem is referenced by:  reurexprg  4625
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