Theorem reurexprg 4640

Description: Convert a restricted existential uniqueness over a pair to a restricted existential quantification and an implication . (Contributed by AV, 3-Apr-2023.)

Hypotheses

Ref	Expression
reuprg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
reuprg.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

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

Step	Hyp	Ref	Expression
1		reuprg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		reuprg.2	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
3	1, 2	reuprg 4639	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))
4	1, 2	rexprg 4632	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
5	4	bicomd 222	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜓 ∨ 𝜒) ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝜑))
6	5	anbi1d 630	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵)) ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))
7	3, 6	bitrd 278	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ∃!wreu 3066  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-v 3434  df-sbc 3717  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by: (None)