MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reuprg Structured version   Visualization version   GIF version

Theorem reuprg 4728
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 4727 . 2 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵)))))
4 orddi 1010 . . 3 (((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵))) ↔ (((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ∧ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))))
5 curryax 892 . . . . . 6 (𝜓 ∨ (𝜓𝐴 = 𝐵))
65biantru 529 . . . . 5 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))))
76bicomi 224 . . . 4 (((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ↔ (𝜓𝜒))
8 curryax 892 . . . . . . . 8 (𝜒 ∨ (𝜒𝐴 = 𝐵))
9 orcom 869 . . . . . . . 8 (((𝜒𝐴 = 𝐵) ∨ 𝜒) ↔ (𝜒 ∨ (𝜒𝐴 = 𝐵)))
108, 9mpbir 231 . . . . . . 7 ((𝜒𝐴 = 𝐵) ∨ 𝜒)
1110biantrur 530 . . . . . 6 (((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)) ↔ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))))
1211bicomi 224 . . . . 5 ((((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))) ↔ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))
13 pm4.79 1004 . . . . 5 (((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)) ↔ ((𝜒𝜓) → 𝐴 = 𝐵))
1412, 13bitri 275 . . . 4 ((((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵))) ↔ ((𝜒𝜓) → 𝐴 = 𝐵))
157, 14anbi12i 627 . . 3 ((((𝜓𝜒) ∧ (𝜓 ∨ (𝜓𝐴 = 𝐵))) ∧ (((𝜒𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒𝐴 = 𝐵) ∨ (𝜓𝐴 = 𝐵)))) ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)))
164, 15bitri 275 . 2 (((𝜓 ∧ (𝜒𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓𝐴 = 𝐵))) ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)))
173, 16bitrdi 287 1 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846   = wceq 1537  wcel 2108  ∃!wreu 3386  {cpr 4650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-v 3490  df-sbc 3805  df-un 3981  df-sn 4649  df-pr 4651
This theorem is referenced by:  reurexprg  4729
  Copyright terms: Public domain W3C validator