Theorem sblimv 1815

Description: Version of sblim 1872 where x and y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)

Hypothesis

Ref	Expression
sblimv.1	|- ( ps -> A. x ps )

Assertion

Ref	Expression
sblimv	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem sblimv

Step	Hyp	Ref	Expression
1		sbimv 1814	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sblimv.1	. . . 4 |- ( ps -> A. x ps )
3	2	sbh 1699	. . 3 |- ( [ y / x ] ps <-> ps )
4	3	imbi2i 224	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) )
5	1, 4	bitri 182	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by: (None)