Theorem seeq2 4841

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 3542	. . 3 |- ( A = B -> B C_ A )
2		sess2 4837	. . 3 |- ( B C_ A -> ( R Se A -> R Se B ) )
3	1, 2	syl 16	. 2 |- ( A = B -> ( R Se A -> R Se B ) )
4		eqimss 3541	. . 3 |- ( A = B -> A C_ B )
5		sess2 4837	. . 3 |- ( A C_ B -> ( R Se B -> R Se A ) )
6	4, 5	syl 16	. 2 |- ( A = B -> ( R Se B -> R Se A ) )
7	3, 6	impbid 191	1 |- ( A = B -> ( R Se A <-> R Se B ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1398   C_ wss 3461   Se wse 4825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-in 3468  df-ss 3475  df-se 4828

This theorem is referenced by:  oieq2  7930