Theorem nfopab2 5176

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 5168	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 2187	. . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfex 2354	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfab 2928	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 4	nfcxfr 2920	1 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812  {cab 2738  Ⅎwnfc 2907  ⟨cop 4590  {copab 5167

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-opab 5168

This theorem is used by:  rexopabb  5499  ssopab2bw  5519  ssopab2b  5521  dmopab  5894  rnopab  5933  funopab  6564  fvopab5  7016  zfrep6OLD  7951  opabdm  33124  opabrn  33125  fpwrelmap  33244  fineqvrep  35701  bj-opabco  38023  vvdifopab  39111  aomclem8  44000  areaquad  44155  modelaxrep  45902  sprsymrelf  48493