Theorem opreu2reu 6328

Description: If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023.) (Revised by AV, 2-Jul-2023.)

Hypothesis

Ref	Expression
opreu2reurex.a	⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
opreu2reu	⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)

Distinct variable groups:   𝐴,𝑎,𝑏,𝑝   𝐵,𝑎,𝑏,𝑝   𝜑,𝑎,𝑏   𝜒,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝜒(𝑎,𝑏)

Proof of Theorem opreu2reu

Step	Hyp	Ref	Expression
1		opreu2reurex.a	. . 3 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))
2	1	opreu2reurex 6327	. 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))
3		2rexreu 3784	. 2 ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)
4	2, 3	sylbi 217	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wrex 3076  ∃!wreu 3386  ⟨cop 4654   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-iun 5017  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by: (None)