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Theorem pairreueq 43549
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 6672 . . . . . 6 (𝑥 = 𝑝 → ((♯‘𝑥) = 2 ↔ (♯‘𝑝) = 2))
2 pairreueq.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
31, 2elrab2 3680 . . . . 5 (𝑝𝑃 ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
43anbi1i 623 . . . 4 ((𝑝𝑃𝜑) ↔ ((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑))
5 anass 469 . . . 4 (((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
64, 5bitri 276 . . 3 ((𝑝𝑃𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
76eubii 2663 . 2 (∃!𝑝(𝑝𝑃𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
8 df-reu 3142 . 2 (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝(𝑝𝑃𝜑))
9 df-reu 3142 . 2 (∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
107, 8, 93bitr4i 304 1 (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  ∃!weu 2646  ∃!wreu 3137  {crab 3139  𝒫 cpw 4535  cfv 6348  2c2 11680  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  requad2  43665
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