Theorem nf2 1598

Description: An alternate definition of df-nf 1390, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)

Assertion

Ref	Expression
nf2	|- ( F/ x ph <-> ( E. x ph -> A. x ph ) )

Proof of Theorem nf2

Step	Hyp	Ref	Expression
1		df-nf 1390	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 1474	. . . 4 |- F/ x A. x ph
3	2	nfri 1452	. . 3 |- ( A. x ph -> A. x A. x ph )
4	3	19.23h 1427	. 2 |- ( A. x ( ph -> A. x ph ) <-> ( E. x ph -> A. x ph ) )
5	1, 4	bitri 182	1 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   F/wnf 1389   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  nf3  1599  nf4dc  1600  nf4r  1601  eusv2i  4205