Theorem soeq2 5055

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 5053	. . . 4 |- ( A C_ B -> ( R Or B -> R Or A ) )
2		soss 5053	. . . 4 |- ( B C_ A -> ( R Or A -> R Or B ) )
3	1, 2	anim12i 590	. . 3 |- ( ( A C_ B /\ B C_ A ) -> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
4		eqss 3618	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		dfbi2 660	. . 3 |- ( ( R Or B <-> R Or A ) <-> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
6	3, 4, 5	3imtr4i 281	. 2 |- ( A = B -> ( R Or B <-> R Or A ) )
7	6	bicomd 213	1 |- ( A = B -> ( R Or A <-> R Or B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-in 3581  df-ss 3588  df-po 5035  df-so 5036

This theorem is referenced by:  weeq2  5103  wemapso2  8458  oemapso  8579  fin2i  9117  isfin2-2  9141  fin1a2lem10  9231  zorn2lem7  9324  zornn0g  9327  opsrtoslem2  19485  sltsolem1  31826  soeq12d  37608  aomclem1  37624