Theorem opreu2reu1 32561

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 32556	. 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 6253	. 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
4	1, 3	bitr4i 278	1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wrex 3061  ∃!wreu 3349  ⟨cop 4587   × cxp 5623  ∃!w2reu 32555

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-iun 4949  df-opab 5162  df-xp 5631  df-rel 5632  df-2reu 32556

This theorem is referenced by: (None)