Theorem nf2 1558

Description: An alternative definition of df-nf 1350, 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 1350	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 1434	. . . 4 |- F/ x A. x ph
3	2	nfri 1412	. . 3 |- ( A. x ph -> A. x A. x ph )
4	3	19.23h 1387	. 2 |- ( A. x ( ph -> A. x ph ) <-> ( E. x ph -> A. x ph ) )
5	1, 4	bitri 173	1 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  nf3  1559  nf4dc  1560  nf4r  1561  eusv2i  4187