Theorem pairreueq 48300

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 6897	. . . . . 6 ⊢ (𝑥 = 𝑝 → ((♯‘𝑥) = 2 ↔ (♯‘𝑝) = 2))
2		pairreueq.p	. . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
3	1, 2	elrab2 3657	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
4	3	anbi1i 636	. . . 4 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝜑) ↔ ((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑))
5		anass 474	. . . 4 ⊢ (((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
6	4, 5	bitri 278	. . 3 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
7	6	eubii 2616	. 2 ⊢ (∃!𝑝(𝑝 ∈ 𝑃 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
8		df-reu 3373	. 2 ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝(𝑝 ∈ 𝑃 ∧ 𝜑))
9		df-reu 3373	. 2 ⊢ (∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑)))
10	7, 8, 9	3bitr4i 306	1 ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!weu 2599  ∃!wreu 3370  {crab 3419  𝒫 cpw 4567  ‘cfv 6543  2c2 12313  ♯chash 14386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-reu 3373  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551

This theorem is used by:  requad2  48429