Theorem sblimv 1906

Description: Version of sblim 1973 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 1905	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sblimv.1	. . . 4 |- ( ps -> A. x ps )
3	2	sbh 1787	. . 3 |- ( [ y / x ] ps <-> ps )
4	3	imbi2i 226	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) )
5	1, 4	bitri 184	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   [wsb 1773

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by: (None)