Theorem sbcor 2990

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 2954	. 2 |- ( [. A / x ]. ( ph \/ ps ) -> A e. _V )
2		sbcex 2954	. . 3 |- ( [. A / x ]. ph -> A e. _V )
3		sbcex 2954	. . 3 |- ( [. A / x ]. ps -> A e. _V )
4	2, 3	jaoi 706	. 2 |- ( ( [. A / x ]. ph \/ [. A / x ]. ps ) -> A e. _V )
5		dfsbcq2 2949	. . 3 |- ( y = A -> ( [ y / x ] ( ph \/ ps ) <-> [. A / x ]. ( ph \/ ps ) ) )
6		dfsbcq2 2949	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
7		dfsbcq2 2949	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
8	6, 7	orbi12d 783	. . 3 |- ( y = A -> ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
9		sbor 1941	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
10	5, 8, 9	vtoclbg 2782	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
11	1, 4, 10	pm5.21nii 694	1 |- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )

Syntax hints:   <-> wb 104   \/ wo 698   = wceq 1342   [wsb 1749   e. wcel 2135   _Vcvv 2721   [.wsbc 2946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-sbc 2947

This theorem is referenced by:  rabrsndc  3638