Theorem soeq2 4294

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 4292	. . . 4 |- ( A C_ B -> ( R Or B -> R Or A ) )
2		soss 4292	. . . 4 |- ( B C_ A -> ( R Or A -> R Or B ) )
3	1, 2	anim12i 336	. . 3 |- ( ( A C_ B /\ B C_ A ) -> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
4		eqss 3157	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		dfbi2 386	. . 3 |- ( ( R Or B <-> R Or A ) <-> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
6	3, 4, 5	3imtr4i 200	. 2 |- ( A = B -> ( R Or B <-> R Or A ) )
7	6	bicomd 140	1 |- ( A = B -> ( R Or A <-> R Or B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   C_ wss 3116   Or wor 4273

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129  df-po 4274  df-iso 4275

This theorem is referenced by: (None)