Theorem nfopab2 5175

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 5167	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 2186	. . . 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 1569  ∃wex 1811  {cab 2738  Ⅎwnfc 2907  ⟨cop 4589  {copab 5166

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-opab 5167

This theorem is used by:  rexopabb  5501  ssopab2bw  5521  ssopab2b  5523  dmopab  5896  rnopab  5935  funopab  6575  fvopab5  7027  zfrep6OLD  7967  opabdm  33205  opabrn  33206  fpwrelmap  33325  fineqvrep  35782  bj-opabco  38110  vvdifopab  39198  aomclem8  44063  areaquad  44218  modelaxrep  45971  sprsymrelf  48577