Theorem nfopab1 4121

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 4114	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 1520	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfab 2354	. 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	1, 3	nfcxfr 2346	1 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516  {cab 2192  Ⅎwnfc 2336  ⟨cop 3641  {copab 4112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-opab 4114

This theorem is referenced by:  nfmpt1  4145  opelopabsb  4314  ssopab2b  4331  dmopab  4898  rnopab  4934  funopab  5315  0neqopab  6003