Theorem nf2 1692

Description: An alternate definition of df-nf 1485, 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 1485	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 1565	. . . 4 |- F/ x A. x ph
3	2	nfri 1543	. . 3 |- ( A. x ph -> A. x A. x ph )
4	3	19.23h 1522	. 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 1371   F/wnf 1484   E.wex 1516

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 1473  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  nf3  1693  nf4dc  1694  nf4r  1695  nfd2  2051  eusv2i  4520