Theorem nf2 1682

Description: An alternate definition of df-nf 1475, 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 1475	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 1555	. . . 4 |- F/ x A. x ph
3	2	nfri 1533	. . 3 |- ( A. x ph -> A. x A. x ph )
4	3	19.23h 1512	. 2 |- ( A. x ( ph -> A. x ph ) <-> ( E. x ph -> A. x ph ) )
5	1, 4	bitri 184	1 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   F/wnf 1474   E.wex 1506

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-gen 1463  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  nf3  1683  nf4dc  1684  nf4r  1685  nfd2  2041  eusv2i  4490