Theorem nfopab1 5189

Description: The first abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
nfopab1	⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem nfopab1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 5182	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 2150	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfab 2904	. 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	1, 3	nfcxfr 2896	1 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779  {cab 2713  Ⅎwnfc 2883  ⟨cop 4607  {copab 5181

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-opab 5182

This theorem is referenced by:  nfmpt1  5220  rexopabb  5503  ssopab2bw  5522  ssopab2b  5524  dmopab  5895  rnopab  5934  funopab  6571  fvopab5  7019  zfrep6  7953  opabdm  32591  opabrn  32592  fpwrelmap  32710  fineqvrep  35126  bj-opabco  37206  vvdifopab  38278  aomclem8  43085  sprsymrelf  47509