Theorem releqd 4518

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 4516	. 2 |- ( A = B -> ( Rel A <-> Rel B ) )
3	1, 2	syl 14	1 |- ( ph -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   Rel wrel 4441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-rel 4443

This theorem is referenced by:  dftpos3  6019  tposfo2  6024  tposf12  6026