Theorem sbhb 2107

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

Assertion

Ref	Expression
sbhb	|- ((ph -> A.xph) <-> 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		nfv 1619	. . . 4 |- F/yph
2	1	sb8 2092	. . 3 |- (A.xph <-> A.y[y / x]ph)
3	2	imbi2i 303	. 2 |- ((ph -> A.xph) <-> (ph -> A.y[y / x]ph))
4		19.21v 1890	. 2 |- (A.y(ph -> [y / x]ph) <-> (ph -> A.y[y / x]ph))
5	3, 4	bitr4i 243	1 |- ((ph -> A.xph) <-> A.y(ph -> [y / x]ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbnf2  2108