Theorem seeq2 4480

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 3303	. . 3 |- ( A = B -> B C_ A )
2		sess2 4478	. . 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 3302	. . 3 |- ( A = B -> A C_ B )
5		sess2 4478	. . 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 1402   C_ wss 3220   Se wse 4469

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233  df-se 4473

This theorem is referenced by: (None)