Theorem reurexprg 4661

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 4660	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))
4	1, 2	rexprg 4654	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
5	4	bicomd 223	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜓 ∨ 𝜒) ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝜑))
6	5	anbi1d 631	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝜓 ∨ 𝜒) ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵)) ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))
7	3, 6	bitrd 279	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒 ∧ 𝜓) → 𝐴 = 𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ∃!wreu 3348  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-v 3442  df-sbc 3741  df-un 3906  df-sn 4581  df-pr 4583

This theorem is referenced by: (None)