Theorem seeq1 4826

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 3523	. . 3 |- ( R = S -> S C_ R )
2		sess1 4822	. . 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 3522	. . 3 |- ( R = S -> R C_ S )
5		sess1 4822	. . 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 193	1 |- ( R = S -> ( R Se A <-> S Se A ) )

Syntax hints:   -> wi 4   <-> wb 187   = wceq 1437   C_ wss 3442   Se wse 4811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-in 3449  df-ss 3456  df-br 4427  df-se 4814

This theorem is referenced by:  oieq1  8027