Theorem reuprg 4636

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 → (𝜑 ↔ 𝜓))
reuprg.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
reuprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 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

Step	Hyp	Ref	Expression
1		reuprg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		reuprg.2	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
3	1, 2	reuprg0 4635	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))))
4		orddi 1006	. . 3 ⊢ (((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))) ↔ (((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ (𝜓 → 𝐴 = 𝐵))) ∧ (((𝜒 → 𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵)))))
5		curryax 890	. . . . . 6 ⊢ (𝜓 ∨ (𝜓 → 𝐴 = 𝐵))
6	5	biantru 529	. . . . 5 ⊢ ((𝜓 ∨ 𝜒) ↔ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ (𝜓 → 𝐴 = 𝐵))))
7	6	bicomi 223	. . . 4 ⊢ (((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ (𝜓 → 𝐴 = 𝐵))) ↔ (𝜓 ∨ 𝜒))
8		curryax 890	. . . . . . . 8 ⊢ (𝜒 ∨ (𝜒 → 𝐴 = 𝐵))
9		orcom 866	. . . . . . . 8 ⊢ (((𝜒 → 𝐴 = 𝐵) ∨ 𝜒) ↔ (𝜒 ∨ (𝜒 → 𝐴 = 𝐵)))
10	8, 9	mpbir 230	. . . . . . 7 ⊢ ((𝜒 → 𝐴 = 𝐵) ∨ 𝜒)
11	10	biantrur 530	. . . . . 6 ⊢ (((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵)) ↔ (((𝜒 → 𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵))))
12	11	bicomi 223	. . . . 5 ⊢ ((((𝜒 → 𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵))) ↔ ((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵)))
13		pm4.79 1000	. . . . 5 ⊢ (((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵)) ↔ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))
14	12, 13	bitri 274	. . . 4 ⊢ ((((𝜒 → 𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵))) ↔ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))
15	7, 14	anbi12i 626	. . 3 ⊢ ((((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ (𝜓 → 𝐴 = 𝐵))) ∧ (((𝜒 → 𝐴 = 𝐵) ∨ 𝜒) ∧ ((𝜒 → 𝐴 = 𝐵) ∨ (𝜓 → 𝐴 = 𝐵)))) ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵)))
16	4, 15	bitri 274	. 2 ⊢ (((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))) ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵)))
17	3, 16	bitrdi 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∃!wreu 3065  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-v 3424  df-sbc 3712  df-un 3888  df-sn 4559  df-pr 4561

This theorem is referenced by:  reurexprg  4637