Theorem seeq1 4317

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 3197	. . 3 |- ( R = S -> S C_ R )
2		sess1 4315	. . 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 3196	. . 3 |- ( R = S -> R C_ S )
5		sess1 4315	. . 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 128	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   C_ wss 3116   Se wse 4307

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-br 3983  df-se 4311

This theorem is referenced by: (None)