Theorem nf2 1555

Description: An alternative definition of df-nf 1347, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)

Assertion

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

Proof of Theorem nf2

Step	Hyp	Ref	Expression
1		df-nf 1347	. 2 |- ( F/xph <-> A.x(ph -> A.xph))
2		nfa1 1431	. . . 4 |- F/xA.xph
3	2	nfri 1409	. . 3 |- (A.xph -> A.xA.xph)
4	3	19.23h 1384	. 2 |- (A.x(ph -> A.xph) <-> (E.xph -> A.xph))
5	1, 4	bitri 173	1 |- ( F/xph <-> (E.xph -> A.xph))

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   F/wnf 1346  E.wex 1378

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 1335  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  nf3  1556  nf4dc  1557  nf4r  1558  eusv2i  4153