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Theorem pairreueq 47497
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 6915 . . . . . 6 (𝑥 = 𝑝 → ((♯‘𝑥) = 2 ↔ (♯‘𝑝) = 2))
2 pairreueq.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
31, 2elrab2 3695 . . . . 5 (𝑝𝑃 ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
43anbi1i 624 . . . 4 ((𝑝𝑃𝜑) ↔ ((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑))
5 anass 468 . . . 4 (((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
64, 5bitri 275 . . 3 ((𝑝𝑃𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
76eubii 2585 . 2 (∃!𝑝(𝑝𝑃𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
8 df-reu 3381 . 2 (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝(𝑝𝑃𝜑))
9 df-reu 3381 . 2 (∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
107, 8, 93bitr4i 303 1 (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!weu 2568  ∃!wreu 3378  {crab 3436  𝒫 cpw 4600  cfv 6561  2c2 12321  chash 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by:  requad2  47610
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