Theorem nfopab1 4074

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

Proof of Theorem nfopab1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 4067	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1496	. . 3 |- F/ x E. x E. y ( z = <. x , y >. /\ ph )
3	2	nfab 2324	. 2 |- F/_ x { z | E. x E. y ( z = <. x , y >. /\ ph ) }
4	1, 3	nfcxfr 2316	1 |- F/_ x { <. x , y >. | ph }

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   {cab 2163   F/_wnfc 2306   <.cop 3597   {copab 4065

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-opab 4067

This theorem is referenced by:  nfmpt1  4098  opelopabsb  4262  ssopab2b  4278  dmopab  4840  rnopab  4876  funopab  5253  0neqopab  5922