Theorem sbimv 1886

Description: Intuitionistic proof of sbim 1946 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 1884	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbi2v 1885	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
3	1, 2	impbii 125	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 104   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by:  sblimv  1887  sbim  1946