Theorem seeq1 4811

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 3471	. . 3 |- ( R = S -> S C_ R )
2		sess1 4807	. . 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 3470	. . 3 |- ( R = S -> R C_ S )
5		sess1 4807	. . 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 195	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   C_ wss 3390   Se wse 4796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-in 3397  df-ss 3404  df-br 4396  df-se 4799

This theorem is referenced by:  oieq1  8045