Theorem nfopab1 4628

Description: The first 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
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 4623	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ φ)}
2		nfe1 1732	. . 3 ⊢ Ⅎx∃x∃y(z = ⟨x, y⟩ ∧ φ)
3	2	nfab 2493	. 2 ⊢ Ⅎx{z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ φ)}
4	1, 3	nfcxfr 2486	1 ⊢ Ⅎx{⟨x, y⟩ ∣ φ}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  Ⅎwnfc 2476  ⟨cop 4561  {copab 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-opab 4623

This theorem is referenced by:  opelopabsb  4697  ssopab2b  4713  dmopab  4915  rnopab  4967  funopab  5139  nfmpt1  5672