Theorem oprabbi 38155

Description: Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Assertion

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

Proof of Theorem oprabbi

Step	Hyp	Ref	Expression
1		eqoprab2b 7460	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓))
2	1	biimpri 228	1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  {coprab 7388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2370  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-oprab 7391

This theorem is referenced by:  mpobi123f  38156