Theorem sbcor 3042

Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
sbcor	|- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )

Proof of Theorem sbcor

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3006	. 2 |- ( [. A / x ]. ( ph \/ ps ) -> A e. _V )
2		sbcex 3006	. . 3 |- ( [. A / x ]. ph -> A e. _V )
3		sbcex 3006	. . 3 |- ( [. A / x ]. ps -> A e. _V )
4	2, 3	jaoi 717	. 2 |- ( ( [. A / x ]. ph \/ [. A / x ]. ps ) -> A e. _V )
5		dfsbcq2 3000	. . 3 |- ( y = A -> ( [ y / x ] ( ph \/ ps ) <-> [. A / x ]. ( ph \/ ps ) ) )
6		dfsbcq2 3000	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
7		dfsbcq2 3000	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
8	6, 7	orbi12d 794	. . 3 |- ( y = A -> ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
9		sbor 1981	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
10	5, 8, 9	vtoclbg 2833	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
11	1, 4, 10	pm5.21nii 705	1 |- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )

Syntax hints:   <-> wb 105   \/ wo 709   = wceq 1372   [wsb 1784   e. wcel 2175   _Vcvv 2771   [.wsbc 2997

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998

This theorem is referenced by:  rabrsndc  3700