Theorem releqd 4695

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 4693	. 2 |- ( A = B -> ( Rel A <-> Rel B ) )
3	1, 2	syl 14	1 |- ( ph -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-rel 4618

This theorem is referenced by:  dftpos3  6241  tposfo2  6246  tposf12  6248  lmreltop  12987  cnprcl2k  13000