Theorem opabid 5502

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker opabidw 5501 when possible. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
opabid	⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Proof of Theorem opabid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5435	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexg 5465	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	bicomd 226	. 2 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4		df-opab 5170	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3638	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ⟨cop 4837  {copab 5169

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-13 2404  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-opab 5170

This theorem is used by:  ssopab2b  5527  brabidga  39446