Theorem seeq2 4270

Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
seeq2	|- ( A = B -> ( R Se A <-> R Se B ) )

Proof of Theorem seeq2

Step	Hyp	Ref	Expression
1		eqimss2 3157	. . 3 |- ( A = B -> B C_ A )
2		sess2 4268	. . 3 |- ( B C_ A -> ( R Se A -> R Se B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( R Se A -> R Se B ) )
4		eqimss 3156	. . 3 |- ( A = B -> A C_ B )
5		sess2 4268	. . 3 |- ( A C_ B -> ( R Se B -> R Se A ) )
6	4, 5	syl 14	. 2 |- ( A = B -> ( R Se B -> R Se A ) )
7	3, 6	impbid 128	1 |- ( A = B -> ( R Se A <-> R Se B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   C_ wss 3076   Se wse 4259

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-in 3082  df-ss 3089  df-se 4263

This theorem is referenced by: (None)