Theorem ralopabb 43868

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 1836	. . 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 320	. . . . 5 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (¬ 𝜓 ↔ ¬ 𝜒))
5	2, 4	rexopabb 5472	. . . 4 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒))
6		annim 405	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑 → 𝜒))
7	6	2exbii 1857	. . . 4 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒))
8	5, 7	bitri 277	. . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒))
9		rexnal 3093	. . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ¬ ∀𝑜 ∈ 𝑂 𝜓)
10	1, 8, 9	3bitr2ri 302	. 2 ⊢ (¬ ∀𝑜 ∈ 𝑂 𝜓 ↔ ¬ ∀𝑥∀𝑦(𝜑 → 𝜒))
11	10	con4bii 323	1 ⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787  ∀wral 3055  ∃wrex 3065  ⟨cop 4563  {copab 5136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5137

This theorem is referenced by: (None)