Theorem nfopab2 5187

Description: The second 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
nfopab2	⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem nfopab2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 5179	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 2192	. . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfex 2364	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfab 2938	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 4	nfcxfr 2930	1 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 400   = wceq 1568  ∃wex 1807  {cab 2748  Ⅎwnfc 2917  ⟨cop 4600  {copab 5178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-opab 5179

This theorem is referenced by:  rexopabb  5516  ssopab2bw  5536  ssopab2b  5538  dmopab  5909  rnopab  5948  funopab  6575  fvopab5  7027  zfrep6OLD  7955  opabdm  32926  opabrn  32927  fpwrelmap  33048  fineqvrep  35485  bj-opabco  37780  vvdifopab  38864  aomclem8  43740  areaquad  43895  modelaxrep  45642  sprsymrelf  48193