Theorem nfopab2 4059

Description: The second abstraction variable in an ordered-pair class abstraction (class builder) 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 4051	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 1489	. . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfex 1630	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfab 2317	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 4	nfcxfr 2309	1 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 103   = wceq 1348  ∃wex 1485  {cab 2156  Ⅎwnfc 2299  ⟨cop 3586  {copab 4049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-opab 4051

This theorem is referenced by:  opelopabsb  4245  ssopab2b  4261  dmopab  4822  rnopab  4858  funopab  5233  0neqopab  5898