Theorem sbhb 1816

Description: Two ways of expressing " x is (effectively) not free in ph." (Contributed by NM, 29-May-2009.)

Assertion

Ref	Expression
sbhb	|- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) )

Distinct variable group:   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem sbhb

Step	Hyp	Ref	Expression
1		ax-17 1419	. . . 4 |- ( ph -> A. y ph )
2	1	sb8h 1734	. . 3 |- ( A. x ph <-> A. y [ y / x ] ph )
3	2	imbi2i 215	. 2 |- ( ( ph -> A. x ph ) <-> ( ph -> A. y [ y / x ] ph ) )
4		19.21v 1753	. 2 |- ( A. y ( ph -> [ y / x ] ph ) <-> ( ph -> A. y [ y / x ] ph ) )
5	3, 4	bitr4i 176	1 |- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbnf2  1857