Theorem soeq2 4053

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 4051	. . . 4 |- ( A C_ B -> ( R Or B -> R Or A ) )
2		soss 4051	. . . 4 |- ( B C_ A -> ( R Or A -> R Or B ) )
3	1, 2	anim12i 321	. . 3 |- ( ( A C_ B /\ B C_ A ) -> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
4		eqss 2960	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		dfbi2 368	. . 3 |- ( ( R Or B <-> R Or A ) <-> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
6	3, 4, 5	3imtr4i 190	. 2 |- ( A = B -> ( R Or B <-> R Or A ) )
7	6	bicomd 129	1 |- ( A = B -> ( R Or A <-> R Or B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   C_ wss 2917   Or wor 4032

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-in 2924  df-ss 2931  df-po 4033  df-iso 4034

This theorem is referenced by: (None)