Theorem seeq1 4806

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 3485	. . 3 |- ( R = S -> S C_ R )
2		sess1 4802	. . 3 |- ( S C_ R -> ( R Se A -> S Se A ) )
3	1, 2	syl 17	. 2 |- ( R = S -> ( R Se A -> S Se A ) )
4		eqimss 3484	. . 3 |- ( R = S -> R C_ S )
5		sess1 4802	. . 3 |- ( R C_ S -> ( S Se A -> R Se A ) )
6	4, 5	syl 17	. 2 |- ( R = S -> ( S Se A -> R Se A ) )
7	3, 6	impbid 194	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 188   = wceq 1444   C_ wss 3404   Se wse 4791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rab 2746  df-v 3047  df-in 3411  df-ss 3418  df-br 4403  df-se 4794

This theorem is referenced by:  oieq1  8027