Theorem releqd 4777

Description: Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)

Hypothesis

Ref	Expression
releqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
releqd	|- ( ph -> ( Rel A <-> Rel B ) )

Proof of Theorem releqd

Step	Hyp	Ref	Expression
1		releqd.1	. 2 |- ( ph -> A = B )
2		releq 4775	. 2 |- ( A = B -> ( Rel A <-> Rel B ) )
3	1, 2	syl 14	1 |- ( ph -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   Rel wrel 4698

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-rel 4700

This theorem is referenced by:  dftpos3  6371  tposfo2  6376  tposf12  6378  imasaddfnlemg  13261  releqgg  13671  dvdsrd  13971  isunitd  13983  lmreltop  14780  cnprcl2k  14793