Theorem nfopab1 5185

Description: The first 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
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 5178	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfe1 2191	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3	2	nfab 2937	. 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	1, 3	nfcxfr 2929	1 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806  {cab 2747  Ⅎwnfc 2916  ⟨cop 4600  {copab 5177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-opab 5178

This theorem is referenced by:  nfmpt1  5214  rexopabb  5513  ssopab2bw  5533  ssopab2b  5535  dmopab  5906  rnopab  5945  funopab  6572  fvopab5  7024  zfrep6OLD  7952  opabdm  32897  opabrn  32898  fpwrelmap  33019  fineqvrep  35450  bj-opabco  37720  vvdifopab  38804  aomclem8  43680  sprsymrelf  48133