Theorem sbim 2235

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 2232	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbi2 2233	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
3	1, 2	impbii 192	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 189   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809

This theorem is referenced by:  sbrim  2236  sblim  2237  sbor  2238  sban  2239  sbbi  2241  sbequ8ALT  2247  sbcimg  3321  mo5f  28176  iuninc  28230  suppss2fOLD  28290  suppss2f  28291  esumpfinvalf  28948  bj-sbnf  31487  wl-sbrimt  31924  wl-sblimt  31925  frege58bcor  36545  frege60b  36547  frege65b  36552  ellimcabssub0  37783