Theorem nfre1 2576

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 2517	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		nfe1 1545	. 2 |- F/ x E. x ( x e. A /\ ph )
3	1, 2	nfxfr 1523	1 |- F/ x E. x e. A ph

Syntax hints:   /\ wa 104   F/wnf 1509   E.wex 1541   e. wcel 2202   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-rex 2517

This theorem is referenced by:  r19.29an  2676  nfiu1  4005  fun11iun  5613  eusvobj2  6014  fodjuomnilemdc  7403  ismkvnex  7414  prarloclem3step  7776  prmuloc2  7847  ltexprlemm  7880  caucvgprprlemaddq  7988  caucvgsrlemgt1  8075  axpre-suploclemres  8181  supinfneg  9890  infsupneg  9891  lbzbi  9911  divalglemeunn  12562  divalglemeuneg  12564  bezoutlemmain  12649  bezout  12662  lss1d  14479  pw1nct  16725  isomninnlem  16762  trirec0  16776  ismkvnnlem  16785