Theorem opreu2reu1 31719

Description: Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)

Hypothesis

Ref	Expression
opreu2reu1.a	⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒 ↔ 𝜑))

Assertion

Ref	Expression
opreu2reu1	⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)

Distinct variable groups:   𝐴,𝑝,𝑥,𝑦   𝐵,𝑝,𝑥,𝑦   𝜒,𝑥,𝑦   𝜑,𝑝,𝑥,𝑦

Allowed substitution hint:   𝜒(𝑝)

Proof of Theorem opreu2reu1

Step	Hyp	Ref	Expression
1		df-2reu 31714	. 2 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
2		opreu2reu1.a	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒 ↔ 𝜑))
3	2	opreu2reurex 6293	. 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
4	1, 3	bitr4i 277	1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wrex 3070  ∃!wreu 3374  ⟨cop 4634   × cxp 5674  ∃!w2reu 31713

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-iun 4999  df-opab 5211  df-xp 5682  df-rel 5683  df-2reu 31714

This theorem is referenced by: (None)