Theorem sbcor 2948

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 2912	. 2 |- ( [. A / x ]. ( ph \/ ps ) -> A e. _V )
2		sbcex 2912	. . 3 |- ( [. A / x ]. ph -> A e. _V )
3		sbcex 2912	. . 3 |- ( [. A / x ]. ps -> A e. _V )
4	2, 3	jaoi 705	. 2 |- ( ( [. A / x ]. ph \/ [. A / x ]. ps ) -> A e. _V )
5		dfsbcq2 2907	. . 3 |- ( y = A -> ( [ y / x ] ( ph \/ ps ) <-> [. A / x ]. ( ph \/ ps ) ) )
6		dfsbcq2 2907	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
7		dfsbcq2 2907	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
8	6, 7	orbi12d 782	. . 3 |- ( y = A -> ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
9		sbor 1925	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
10	5, 8, 9	vtoclbg 2742	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
11	1, 4, 10	pm5.21nii 693	1 |- ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )

Syntax hints:   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   [wsb 1735   _Vcvv 2681   [.wsbc 2904

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by:  rabrsndc  3586