Theorem sbcor 2858

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 2823	. 2 |- ( [. A / x ]. ( ph \/ ps ) -> A e. _V )
2		sbcex 2823	. . 3 |- ( [. A / x ]. ph -> A e. _V )
3		sbcex 2823	. . 3 |- ( [. A / x ]. ps -> A e. _V )
4	2, 3	jaoi 668	. 2 |- ( ( [. A / x ]. ph \/ [. A / x ]. ps ) -> A e. _V )
5		dfsbcq2 2818	. . 3 |- ( y = A -> ( [ y / x ] ( ph \/ ps ) <-> [. A / x ]. ( ph \/ ps ) ) )
6		dfsbcq2 2818	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
7		dfsbcq2 2818	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
8	6, 7	orbi12d 739	. . 3 |- ( y = A -> ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
9		sbor 1869	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
10	5, 8, 9	vtoclbg 2659	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
11	1, 4, 10	pm5.21nii 652	1 |- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )

Syntax hints:   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   [wsb 1685   _Vcvv 2601   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  rabrsndc  3460