Theorem nfopab2 4201

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	|- F/_ y { <. x , y >. | ph }

Proof of Theorem nfopab2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 4193	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1549	. . . 4 |- F/ y E. y ( z = <. x , y >. /\ ph )
3	2	nfex 1690	. . 3 |- F/ y E. x E. y ( z = <. x , y >. /\ ph )
4	3	nfab 2397	. 2 |- F/_ y { z | E. x E. y ( z = <. x , y >. /\ ph ) }
5	1, 4	nfcxfr 2389	1 |- F/_ y { <. x , y >. | ph }

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   E.wex 1545   {cab 2224   F/_wnfc 2379   <.cop 3712   {copab 4191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-opab 4193

This theorem is used by:  opelopabsb  4402  ssopab2b  4419  dmopab  4992  rnopab  5029  funopab  5412  0neqopab  6133