Theorem releqd 4839

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 4837	. 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 1398   Rel wrel 4759

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227  df-rel 4761

This theorem is referenced by:  dftpos3  6506  tposfo2  6511  tposf12  6513  imasaddfnlemg  13578  releqgg  13973  dvdsrd  14339  isunitd  14351  cnprcl2k  15197