Theorem opreu2reu 6291

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 6290	. 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))
3		2rexreu 3720	. 2 ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)
4	2, 3	sylbi 220	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wrex 3087  ∃!wreu 3364  ⟨cop 4590   × cxp 5649

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-opab 5168  df-xp 5657  df-rel 5658

This theorem is used by: (None)