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Theorem reurexprg 4632
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 → (𝜑𝜓))
reuprg.2 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
reurexprg ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((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 reurexprg
StepHypRef Expression
1 reuprg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
2 reuprg.2 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
31, 2reuprg 4631 . 2 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
41, 2rexprg 4625 . . . 4 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
54bicomd 225 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝜓𝜒) ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝜑))
65anbi1d 631 . 2 ((𝐴𝑉𝐵𝑊) → (((𝜓𝜒) ∧ ((𝜒𝜓) → 𝐴 = 𝐵)) ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
73, 6bitrd 281 1 ((𝐴𝑉𝐵𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ((𝜒𝜓) → 𝐴 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1530  wcel 2107  wrex 3137  ∃!wreu 3138  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-reu 3143  df-v 3495  df-sbc 3771  df-un 3939  df-sn 4560  df-pr 4562
This theorem is referenced by: (None)
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