Theorem sbimv 1773

Description: Intuitionistic proof of sbim 1827 where x and y are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem sbimv

Step	Hyp	Ref	Expression
1		sbi1v 1771	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbi2v 1772	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
3	1, 2	impbii 117	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 98   [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-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-sb 1646

This theorem is referenced by:  sblimv  1774  sbim  1827