Theorem seeq1 4851

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 3557	. . 3 |- ( R = S -> S C_ R )
2		sess1 4847	. . 3 |- ( S C_ R -> ( R Se A -> S Se A ) )
3	1, 2	syl 16	. 2 |- ( R = S -> ( R Se A -> S Se A ) )
4		eqimss 3556	. . 3 |- ( R = S -> R C_ S )
5		sess1 4847	. . 3 |- ( R C_ S -> ( S Se A -> R Se A ) )
6	4, 5	syl 16	. 2 |- ( R = S -> ( S Se A -> R Se A ) )
7	3, 6	impbid 191	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   C_ wss 3476   Se wse 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-br 4448  df-se 4839

This theorem is referenced by:  oieq1  7938