Theorem sbi2 2393

Description: Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)

Assertion

Ref	Expression
sbi2	|- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )

Proof of Theorem sbi2

Step	Hyp	Ref	Expression
1		sbn 2391	. . 3 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
2		pm2.21 120	. . . 4 |- ( -. ph -> ( ph -> ps ) )
3	2	sbimi 1886	. . 3 |- ( [ y / x ] -. ph -> [ y / x ] ( ph -> ps ) )
4	1, 3	sylbir 225	. 2 |- ( -. [ y / x ] ph -> [ y / x ] ( ph -> ps ) )
5		ax-1 6	. . 3 |- ( ps -> ( ph -> ps ) )
6	5	sbimi 1886	. 2 |- ( [ y / x ] ps -> [ y / x ] ( ph -> ps ) )
7	4, 6	ja 173	1 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   [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:  sbim  2395