Theorem seeq1 4386

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 3248	. . 3 |- ( R = S -> S C_ R )
2		sess1 4384	. . 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 3247	. . 3 |- ( R = S -> R C_ S )
5		sess1 4384	. . 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 129	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   C_ wss 3166   Se wse 4376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4162

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179  df-br 4045  df-se 4380

This theorem is referenced by: (None)