Theorem feq23d 5524

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 2239	. 2 |- ( ph -> F = F )
2		feq23d.1	. 2 |- ( ph -> A = C )
3		feq23d.2	. 2 |- ( ph -> B = D )
4	1, 2, 3	feq123d 5519	1 |- ( ph -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   -->wf 5368

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  intopsn  13664  mhmpropd  13750  grp1inv  13889  isrhm2d  14445  rhmopp  14456