Theorem nfre1 2509

Description: x is not free in E. x e. A ph. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)

Assertion

Ref	Expression
nfre1	|- F/ x E. x e. A ph

Proof of Theorem nfre1

Step	Hyp	Ref	Expression
1		df-rex 2450	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		nfe1 1484	. 2 |- F/ x E. x ( x e. A /\ ph )
3	1, 2	nfxfr 1462	1 |- F/ x E. x e. A ph

Syntax hints:   /\ wa 103   F/wnf 1448   E.wex 1480   e. wcel 2136   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-rex 2450

This theorem is referenced by:  r19.29an  2608  nfiu1  3896  fun11iun  5453  eusvobj2  5828  fodjuomnilemdc  7108  ismkvnex  7119  prarloclem3step  7437  prmuloc2  7508  ltexprlemm  7541  caucvgprprlemaddq  7649  caucvgsrlemgt1  7736  axpre-suploclemres  7842  supinfneg  9533  infsupneg  9534  lbzbi  9554  divalglemeunn  11858  divalglemeuneg  11860  bezoutlemmain  11931  bezout  11944  pw1nct  13883  isomninnlem  13909  trirec0  13923  ismkvnnlem  13931