Theorem sbor 2398

Description: Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)

Assertion

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

Proof of Theorem sbor

Step	Hyp	Ref	Expression
1		sbim 2395	. . 3 |- ( [ y / x ] ( -. ph -> ps ) <-> ( [ y / x ] -. ph -> [ y / x ] ps ) )
2		sbn 2391	. . . 4 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
3	2	imbi1i 339	. . 3 |- ( ( [ y / x ] -. ph -> [ y / x ] ps ) <-> ( -. [ y / x ] ph -> [ y / x ] ps ) )
4	1, 3	bitri 264	. 2 |- ( [ y / x ] ( -. ph -> ps ) <-> ( -. [ y / x ] ph -> [ y / x ] ps ) )
5		df-or 385	. . 3 |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
6	5	sbbii 1887	. 2 |- ( [ y / x ] ( ph \/ ps ) <-> [ y / x ] ( -. ph -> ps ) )
7		df-or 385	. 2 |- ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( -. [ y / x ] ph -> [ y / x ] ps ) )
8	4, 6, 7	3bitr4i 292	1 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   [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:  sbcor  3479  unab  3894  sbcorgOLD  38740