Theorem nfopab1 5170

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 5163	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 2156	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfab 2905	. 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	1, 3	nfcxfr 2897	1 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  Ⅎwnfc 2884  ⟨cop 4588  {copab 5162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-opab 5163

This theorem is referenced by:  nfmpt1  5199  rexopabb  5484  ssopab2bw  5503  ssopab2b  5505  dmopab  5872  rnopab  5911  funopab  6535  fvopab5  6983  zfrep6  7909  opabdm  32701  opabrn  32702  fpwrelmap  32823  fineqvrep  35292  bj-opabco  37443  vvdifopab  38516  aomclem8  43418  sprsymrelf  47855