Theorem reurexprg 4662

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 4661	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))
4	1, 2	rexprg 4655	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
5	4	bicomd 225	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜓 ∨ 𝜒) ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝜑))
6	5	anbi1d 640	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵)) ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))
7	3, 6	bitrd 281	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ∃!wreu 3364  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-v 3455  df-sbc 3745  df-un 3909  df-sn 4582  df-pr 4584

This theorem is referenced by: (None)