Theorem eqoprab2b 7417

Description: Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5490. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker eqoprab2bw 7416 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 7415	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓))
2		ssoprab2b 7415	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))
3	1, 2	anbi12i 628	. 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 3945	. 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 1891	. . . 4 ⊢ (∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)))
6	5	albii 1820	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)))
7		19.26 1871	. . 3 ⊢ (∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)))
8	6, 7	bitri 275	. 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)))
9	3, 4, 8	3bitr4i 303	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ⊆ wss 3897  {coprab 7347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-oprab 7350

This theorem is referenced by:  oprabbi  38209