Theorem pairreueq 44962

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	⊢ (∃!𝑝 ∈ 𝑃 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

Step	Hyp	Ref	Expression
1		fveqeq2 6783	. . . . . 6 ⊢ (𝑥 = 𝑝 → ((♯‘𝑥) = 2 ↔ (♯‘𝑝) = 2))
2		pairreueq.p	. . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
3	1, 2	elrab2 3627	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
4	3	anbi1i 624	. . . 4 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝜑) ↔ ((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑))
5		anass 469	. . . 4 ⊢ (((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
6	4, 5	bitri 274	. . 3 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
7	6	eubii 2585	. 2 ⊢ (∃!𝑝(𝑝 ∈ 𝑃 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
8		df-reu 3072	. 2 ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝(𝑝 ∈ 𝑃 ∧ 𝜑))
9		df-reu 3072	. 2 ⊢ (∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
10	7, 8, 9	3bitr4i 303	1 ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∃!wreu 3066  {crab 3068  𝒫 cpw 4533  ‘cfv 6433  2c2 12028  ♯chash 14044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by:  requad2  45075