Theorem sbim 2114

Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbim

Step	Hyp	Ref	Expression
1		sbi1 2112	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbi2 2113	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
3	1, 2	impbii 181	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 177   [wsb 1655

This theorem is referenced by:  sbor  2115  sbrim  2116  sblim  2117  sban  2118  sbbi  2120  sbequ8  2128  sbcimg  3162  mo5f  23925  iuninc  23964  suppss2f  24002  esumpfinvalf  24419

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656