Theorem seeq2 4405

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 3256	. . 3 |- ( A = B -> B C_ A )
2		sess2 4403	. . 3 |- ( B C_ A -> ( R Se A -> R Se B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( R Se A -> R Se B ) )
4		eqimss 3255	. . 3 |- ( A = B -> A C_ B )
5		sess2 4403	. . 3 |- ( A C_ B -> ( R Se B -> R Se A ) )
6	4, 5	syl 14	. 2 |- ( A = B -> ( R Se B -> R Se A ) )
7	3, 6	impbid 129	1 |- ( A = B -> ( R Se A <-> R Se B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   C_ wss 3174   Se wse 4394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-sep 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-in 3180  df-ss 3187  df-se 4398

This theorem is referenced by: (None)