Theorem soeq2 4238

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 4236	. . . 4 |- ( A C_ B -> ( R Or B -> R Or A ) )
2		soss 4236	. . . 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 3112	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		dfbi2 385	. . 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 1331   C_ wss 3071   Or wor 4217

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-in 3077  df-ss 3084  df-po 4218  df-iso 4219

This theorem is referenced by: (None)