Theorem releq 4710

Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
releq	|- ( A = B -> ( Rel A <-> Rel B ) )

Proof of Theorem releq

Step	Hyp	Ref	Expression
1		sseq1 3180	. 2 |- ( A = B -> ( A C_ ( _V X. _V ) <-> B C_ ( _V X. _V ) ) )
2		df-rel 4635	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4635	. 2 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	1, 2, 3	3bitr4g 223	1 |- ( A = B -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   _Vcvv 2739   C_ wss 3131   X. cxp 4626   Rel wrel 4633

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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-rel 4635

This theorem is referenced by:  releqi  4711  releqd  4712  dfrel2  5081  tposfn2  6269  ereq1  6544