Theorem rb-ax1 1677

Description: The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rb-ax1	|- ( -. ( -. ps \/ ch ) \/ ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) )

Proof of Theorem rb-ax1

Step	Hyp	Ref	Expression
1		orim2 886	. . 3 |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
2		imor 428	. . 3 |- ( ( ps -> ch ) <-> ( -. ps \/ ch ) )
3		imor 428	. . 3 |- ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) <-> ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) )
4	1, 2, 3	3imtr3i 280	. 2 |- ( ( -. ps \/ ch ) -> ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) )
5	4	imori 429	1 |- ( -. ( -. ps \/ ch ) \/ ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386

This theorem is referenced by:  rbsyl  1681  rblem1  1682  rblem2  1683  rblem4  1685  re2luk1  1690