Theorem soeq2 4363

Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)

Assertion

Ref	Expression
soeq2	|- ( A = B -> ( R Or A <-> R Or B ) )

Proof of Theorem soeq2

Step	Hyp	Ref	Expression
1		soss 4361	. . . 4 |- ( A C_ B -> ( R Or B -> R Or A ) )
2		soss 4361	. . . 4 |- ( B C_ A -> ( R Or A -> R Or B ) )
3	1, 2	anim12i 338	. . 3 |- ( ( A C_ B /\ B C_ A ) -> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
4		eqss 3208	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		dfbi2 388	. . 3 |- ( ( R Or B <-> R Or A ) <-> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
6	3, 4, 5	3imtr4i 201	. 2 |- ( A = B -> ( R Or B <-> R Or A ) )
7	6	bicomd 141	1 |- ( A = B -> ( R Or A <-> R Or B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   C_ wss 3166   Or wor 4342

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-in 3172  df-ss 3179  df-po 4343  df-iso 4344

This theorem is referenced by: (None)