Theorem eqoprab2b 7324

Description: Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5458. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker eqoprab2bw 7323 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.)

Assertion

Ref	Expression
eqoprab2b	⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓))

Proof of Theorem eqoprab2b

Step	Hyp	Ref	Expression
1		ssoprab2b 7322	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓))
2		ssoprab2b 7322	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))
3	1, 2	anbi12i 626	. 2 ⊢ (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)))
4		eqss 3932	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}))
5		2albiim 1894	. . . 4 ⊢ (∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)))
6	5	albii 1823	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)))
7		19.26 1874	. . 3 ⊢ (∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)))
8	6, 7	bitri 274	. 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)))
9	3, 4, 8	3bitr4i 302	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ⊆ wss 3883  {coprab 7256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-oprab 7259

This theorem is referenced by:  oprabbi  36246