Theorem seeq2 4784

Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
seeq2	|- ( A = B -> ( R Se A <-> R Se B ) )

Proof of Theorem seeq2

Step	Hyp	Ref	Expression
1		eqimss2 3452	. . 3 |- ( A = B -> B C_ A )
2		sess2 4780	. . 3 |- ( B C_ A -> ( R Se A -> R Se B ) )
3	1, 2	syl 17	. 2 |- ( A = B -> ( R Se A -> R Se B ) )
4		eqimss 3451	. . 3 |- ( A = B -> A C_ B )
5		sess2 4780	. . 3 |- ( A C_ B -> ( R Se B -> R Se A ) )
6	4, 5	syl 17	. 2 |- ( A = B -> ( R Se B -> R Se A ) )
7	3, 6	impbid 195	1 |- ( A = B -> ( R Se A <-> R Se B ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1447   C_ wss 3371   Se wse 4768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2741  df-rab 2745  df-v 3014  df-in 3378  df-ss 3385  df-se 4771

This theorem is referenced by:  oieq2  8014