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Theorem pairreueq 44024
Description: Two equivalent representations of the existence of a unique proper pair. (Contributed by AV, 1-Mar-2023.)
Hypothesis
Ref Expression
pairreueq.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
Assertion
Ref Expression
pairreueq (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))
Distinct variable groups:   𝑥,𝑝   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥,𝑝)   𝑃(𝑥,𝑝)   𝑉(𝑝)

Proof of Theorem pairreueq
StepHypRef Expression
1 fveqeq2 6658 . . . . . 6 (𝑥 = 𝑝 → ((♯‘𝑥) = 2 ↔ (♯‘𝑝) = 2))
2 pairreueq.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
31, 2elrab2 3634 . . . . 5 (𝑝𝑃 ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
43anbi1i 626 . . . 4 ((𝑝𝑃𝜑) ↔ ((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑))
5 anass 472 . . . 4 (((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
64, 5bitri 278 . . 3 ((𝑝𝑃𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
76eubii 2648 . 2 (∃!𝑝(𝑝𝑃𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
8 df-reu 3116 . 2 (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝(𝑝𝑃𝜑))
9 df-reu 3116 . 2 (∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
107, 8, 93bitr4i 306 1 (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  ∃!weu 2631  ∃!wreu 3111  {crab 3113  𝒫 cpw 4500  cfv 6328  2c2 11684  chash 13690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-reu 3116  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336
This theorem is referenced by:  requad2  44138
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