Theorem ralopabb 44165

Description: Restricted universal quantification over an ordered-pair class abstraction. (Contributed by RP, 25-Sep-2024.)

Hypotheses

Ref	Expression
ralopabb.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
ralopabb.p	⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralopabb	⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒))

Distinct variable groups:   𝑜,𝑂   𝑥,𝑜,𝑦   𝜑,𝑜   𝜓,𝑥,𝑦   𝜒,𝑜

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑜)   𝜒(𝑥, 𝑦)   𝑂(𝑥, 𝑦)

Proof of Theorem ralopabb

Step	Hyp	Ref	Expression
1		2nalexn 1857	. . 3 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒))
2		ralopabb.o	. . . . 5 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3		ralopabb.p	. . . . . 6 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))
4	3	notbid 321	. . . . 5 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (¬ 𝜓 ↔ ¬ 𝜒))
5	2, 4	rexopabb 5511	. . . 4 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒))
6		annim 408	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑 → 𝜒))
7	6	2exbii 1878	. . . 4 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒))
8	5, 7	bitri 278	. . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒))
9		rexnal 3116	. . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ¬ ∀𝑜 ∈ 𝑂 𝜓)
10	1, 8, 9	3bitr2ri 303	. 2 ⊢ (¬ ∀𝑜 ∈ 𝑂 𝜓 ↔ ¬ ∀𝑥∀𝑦(𝜑 → 𝜒))
11	10	con4bii 324	1 ⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567   = wceq 1569  ∃wex 1808  ∀wral 3078  ∃wrex 3088  ⟨cop 4594  {copab 5172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173

This theorem is used by: (None)