Theorem sbim 2395

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 2392	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbi2 2393	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )
3	1, 2	impbii 199	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 196   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbrim  2396  sblim  2397  sbor  2398  sban  2399  sbbi  2401  sbequ8ALT  2407  sbcimg  3477  mo5f  29324  iuninc  29379  suppss2f  29439  esumpfinvalf  30138  bj-sbnf  32828  wl-sbrimt  33331  wl-sblimt  33332  frege58bcor  38197  frege60b  38199  frege65b  38204  ellimcabssub0  39849