Theorem sbim 2193

Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 14-May-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 2190	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbi2 2191	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
3	1, 2	impbii 190	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 187   [wsb 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791

This theorem is referenced by:  sbrim  2194  sblim  2195  sbor  2196  sban  2197  sbbi  2199  sbequ8ALT  2205  sbcimg  3341  mo5f  28118  iuninc  28178  suppss2fOLD  28239  suppss2f  28240  esumpfinvalf  28905  bj-sbnf  31411  wl-sbrimt  31842  wl-sblimt  31843  frege58bcor  36469  frege60b  36471  frege65b  36476  ellimcabssub0  37637