Theorem opreu2reu1 32685

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 32680	. 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 6283	. 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
4	1, 3	bitr4i 280	1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wrex 3088  ∃!wreu 3367  ⟨cop 4590   × cxp 5647  ∃!w2reu 32679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-opab 5165  df-xp 5655  df-rel 5656  df-2reu 32680

This theorem is referenced by: (None)