Theorem nf2 1614

Description: An alternate definition of df-nf 1405, 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 1405	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 1489	. . . 4 |- F/ x A. x ph
3	2	nfri 1467	. . 3 |- ( A. x ph -> A. x A. x ph )
4	3	19.23h 1442	. 2 |- ( A. x ( ph -> A. x ph ) <-> ( E. x ph -> A. x ph ) )
5	1, 4	bitri 183	1 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1297   F/wnf 1404   E.wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1393  ax-ie2 1438  ax-4 1455  ax-ial 1482

This theorem depends on definitions:  df-bi 116  df-nf 1405

This theorem is referenced by:  nf3  1615  nf4dc  1616  nf4r  1617  eusv2i  4314