Theorem sbor 1982

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 1914	. . . 4 |- ( [ z / x ] ( ph \/ ps ) <-> ( [ z / x ] ph \/ [ z / x ] ps ) )
2	1	sbbii 1788	. . 3 |- ( [ y / z ] [ z / x ] ( ph \/ ps ) <-> [ y / z ] ( [ z / x ] ph \/ [ z / x ] ps ) )
3		sborv 1914	. . 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 1549	. . 3 |- ( ( ph \/ ps ) -> A. z ( ph \/ ps ) )
6	5	sbco2vh 1973	. 2 |- ( [ y / z ] [ z / x ] ( ph \/ ps ) <-> [ y / x ] ( ph \/ ps ) )
7		ax-17 1549	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1973	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1549	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1973	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	orbi12i 766	. 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 710   [wsb 1785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbcor  3043  sbcorg  3044  unab  3440