Theorem sbc3orgOLD 38742

Description: sbcorgOLD 38740 with a 3-disjuncts. This proof is sbc3orgVD 39086 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbc3orgOLD	|- ( A e. V -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )

Proof of Theorem sbc3orgOLD

Step	Hyp	Ref	Expression
1		sbcorgOLD 38740	. . 3 |- ( A e. V -> ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) )
2		df-3or 1038	. . . . 5 |- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
3	2	bicomi 214	. . . 4 |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
4	3	sbcbiiOLD 38741	. . 3 |- ( A e. V -> ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) )
5		sbcorgOLD 38740	. . . 4 |- ( A e. V -> ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) )
6	5	orbi1d 739	. . 3 |- ( A e. V -> ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) )
7	1, 4, 6	3bitr3d 298	. 2 |- ( A e. V -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) )
8		df-3or 1038	. 2 |- ( ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) )
9	7, 8	syl6bbr 278	1 |- ( A e. V -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   \/ w3o 1036   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcoreleleqVD  39095