Theorem sbor 1973

Description: Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)

Assertion

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

Proof of Theorem sbor

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sborv 1905	. . . 4 |- ( [ z / x ] ( ph \/ ps ) <-> ( [ z / x ] ph \/ [ z / x ] ps ) )
2	1	sbbii 1779	. . 3 |- ( [ y / z ] [ z / x ] ( ph \/ ps ) <-> [ y / z ] ( [ z / x ] ph \/ [ z / x ] ps ) )
3		sborv 1905	. . 3 |- ( [ y / z ] ( [ z / x ] ph \/ [ z / x ] ps ) <-> ( [ y / z ] [ z / x ] ph \/ [ y / z ] [ z / x ] ps ) )
4	2, 3	bitri 184	. 2 |- ( [ y / z ] [ z / x ] ( ph \/ ps ) <-> ( [ y / z ] [ z / x ] ph \/ [ y / z ] [ z / x ] ps ) )
5		ax-17 1540	. . 3 |- ( ( ph \/ ps ) -> A. z ( ph \/ ps ) )
6	5	sbco2vh 1964	. 2 |- ( [ y / z ] [ z / x ] ( ph \/ ps ) <-> [ y / x ] ( ph \/ ps ) )
7		ax-17 1540	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1964	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1540	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1964	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	orbi12i 765	. 2 |- ( ( [ y / z ] [ z / x ] ph \/ [ y / z ] [ z / x ] ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
12	4, 6, 11	3bitr3i 210	1 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )

Syntax hints:   <-> wb 105   \/ wo 709   [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  sbcor  3034  sbcorg  3035  unab  3430