Theorem nfre1 2575

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 2516	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		nfe1 1544	. 2 |- F/ x E. x ( x e. A /\ ph )
3	1, 2	nfxfr 1522	1 |- F/ x E. x e. A ph

Syntax hints:   /\ wa 104   F/wnf 1508   E.wex 1540   e. wcel 2202   E.wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-rex 2516

This theorem is referenced by:  r19.29an  2675  nfiu1  4000  fun11iun  5604  eusvobj2  6004  fodjuomnilemdc  7343  ismkvnex  7354  prarloclem3step  7716  prmuloc2  7787  ltexprlemm  7820  caucvgprprlemaddq  7928  caucvgsrlemgt1  8015  axpre-suploclemres  8121  supinfneg  9829  infsupneg  9830  lbzbi  9850  divalglemeunn  12500  divalglemeuneg  12502  bezoutlemmain  12587  bezout  12600  lss1d  14416  pw1nct  16655  isomninnlem  16685  trirec0  16699  ismkvnnlem  16708