Theorem sbcorg 3008

Description: Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)

Assertion

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

Proof of Theorem sbcorg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2965	. 2 |- ( y = A -> ( [ y / x ] ( ph \/ ps ) <-> [. A / x ]. ( ph \/ ps ) ) )
2		dfsbcq2 2965	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2965	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	orbi12d 793	. 2 |- ( y = A -> ( ( [ y / x ] ph \/ [ y / x ] ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
5		sbor 1954	. 2 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2798	1 |- ( A e. V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   = wceq 1353   [wsb 1762   e. wcel 2148   [.wsbc 2962

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963

This theorem is referenced by: (None)