Theorem poeq2 4055

Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
poeq2	|- ( A = B -> ( R Po A <-> R Po B ) )

Proof of Theorem poeq2

Step	Hyp	Ref	Expression
1		eqimss2 3052	. . 3 |- ( A = B -> B C_ A )
2		poss 4053	. . 3 |- ( B C_ A -> ( R Po A -> R Po B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( R Po A -> R Po B ) )
4		eqimss 3051	. . 3 |- ( A = B -> A C_ B )
5		poss 4053	. . 3 |- ( A C_ B -> ( R Po B -> R Po A ) )
6	4, 5	syl 14	. 2 |- ( A = B -> ( R Po B -> R Po A ) )
7	3, 6	impbid 127	1 |- ( A = B -> ( R Po A <-> R Po B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   C_ wss 2973   Po wpo 4049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-in 2979  df-ss 2986  df-po 4051

This theorem is referenced by: (None)