Theorem seeq1 4123

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 3062	. . 3 |- ( R = S -> S C_ R )
2		sess1 4121	. . 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 3061	. . 3 |- ( R = S -> R C_ S )
5		sess1 4121	. . 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 1285   C_ wss 2983   Se wse 4113

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2612  df-in 2989  df-ss 2996  df-br 3807  df-se 4117

This theorem is referenced by: (None)