Theorem opabbi 35943

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

Assertion

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

Proof of Theorem opabbi

Step	Hyp	Ref	Expression
1		eqopab2b 5408	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓))
2	1	biimpri 231	1 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540   = wceq 1542  {copab 5093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-13 2371  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-v 3400  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-opab 5094

This theorem is referenced by: (None)