Theorem seeq1 4094

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

Assertion

Ref	Expression
seeq1	|- ( R = S -> ( R Se A <-> S Se A ) )

Proof of Theorem seeq1

Step	Hyp	Ref	Expression
1		eqimss2 3052	. . 3 |- ( R = S -> S C_ R )
2		sess1 4092	. . 3 |- ( S C_ R -> ( R Se A -> S Se A ) )
3	1, 2	syl 14	. 2 |- ( R = S -> ( R Se A -> S Se A ) )
4		eqimss 3051	. . 3 |- ( R = S -> R C_ S )
5		sess1 4092	. . 3 |- ( R C_ S -> ( S Se A -> R Se A ) )
6	4, 5	syl 14	. 2 |- ( R = S -> ( S Se A -> R Se A ) )
7	3, 6	impbid 127	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   C_ wss 2973   Se wse 4084

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-br 3786  df-se 4088

This theorem is referenced by: (None)