Theorem nfex 1648

Description: If x is not free in ph, it is not free in E. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
nfex.1	|- F/ x ph

Assertion

Ref	Expression
nfex	|- F/ x E. y ph

Proof of Theorem nfex

Step	Hyp	Ref	Expression
1		nfex.1	. . . 4 |- F/ x ph
2	1	nfri 1530	. . 3 |- ( ph -> A. x ph )
3	2	hbex 1647	. 2 |- ( E. y ph -> A. x E. y ph )
4	3	nfi 1473	1 |- F/ x E. y ph

Syntax hints:   F/wnf 1471   E.wex 1503

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  eeor  1706  cbvexv1  1763  cbvex2  1934  eean  1947  nfsbv  1963  nfeu1  2053  nfeuv  2060  nfel  2345  ceqsex2  2801  nfopab  4098  nfopab2  4100  cbvopab1  4103  cbvopab1s  4105  repizf2  4192  copsex2t  4275  copsex2g  4276  euotd  4284  onintrab2im  4551  mosubopt  4725  nfco  4828  dfdmf  4856  dfrnf  4904  nfdm  4907  fv3  5578  nfoprab2  5969  nfoprab3  5970  nfoprab  5971  cbvoprab1  5991  cbvoprab2  5992  cbvoprab3  5995  cnvoprab  6289  ac6sfi  6956  cc3  7330  nfsum1  11502  nfsum  11503  fsum2dlemstep  11580  nfcprod1  11700  nfcprod  11701  fprod2dlemstep  11768  lss1d  13882