Theorem releq 3243

Description: Equality theorem for the relation predicate.

Assertion

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

Proof of Theorem releq

Step	Hyp	Ref	Expression
1		sseq1 2082	. 2 |- (A = B -> (A (_ (V X. V) <-> B (_ (V X. V)))
2		df-rel 3185	. 2 |- (Rel A <-> A (_ (V X. V))
3		df-rel 3185	. 2 |- (Rel B <-> B (_ (V X. V))
4	1, 2, 3	3bitr4g 555	1 |- (A = B -> (Rel A <-> Rel B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956  Vcvv 1811   (_ wss 2047   X. cxp 3168  Rel wrel 3175

This theorem is referenced by:  releqi 3244  dfrel2 3485  tfrlem6 3916  metne0 7821  lmrel 7927  isps 8645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-rel 3185

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