Theorem nex 1553

Description: Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)

Hypothesis

Ref	Expression
nex.1	|- -. ph

Assertion

Ref	Expression
nex	|- -. E. x ph

Proof of Theorem nex

Step	Hyp	Ref	Expression
1		alnex 1552	. 2 |- ( A. x -. ph <-> -. E. x ph )
2		nex.1	. 2 |- -. ph
3	1, 2	mpgbi 1505	1 |- -. E. x ph

This proof depends on syntax axioms:   -. wn 3   E.wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is used by:  ru  3050  0nelxp  4802  0xp  4855  dm0  4995  co02  5301  0fv  5734  mpo0  6158  0npr  7850  0g0  13696  gzsum0  13713