Theorem feq23d 5333

Description: Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)

Hypotheses

Ref	Expression
feq23d.1	|- ( ph -> A = C )
feq23d.2	|- ( ph -> B = D )

Assertion

Ref	Expression
feq23d	|- ( ph -> ( F : A --> B <-> F : C --> D ) )

Proof of Theorem feq23d

Step	Hyp	Ref	Expression
1		eqidd 2166	. 2 |- ( ph -> F = F )
2		feq23d.1	. 2 |- ( ph -> A = C )
3		feq23d.2	. 2 |- ( ph -> B = D )
4	1, 2, 3	feq123d 5328	1 |- ( ph -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   -->wf 5184

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  intopsn  12598