Theorem poeq2 4403

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 3283	. . 3 |- ( A = B -> B C_ A )
2		poss 4401	. . 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 3282	. . 3 |- ( A = B -> A C_ B )
5		poss 4401	. . 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 129	1 |- ( A = B -> ( R Po A <-> R Po B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   C_ wss 3201   Po wpo 4397

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2516  df-in 3207  df-ss 3214  df-po 4399

This theorem is referenced by: (None)