Theorem feq123d 5398

Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
feq12d.1	|- ( ph -> F = G )
feq12d.2	|- ( ph -> A = B )
feq123d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
feq123d	|- ( ph -> ( F : A --> C <-> G : B --> D ) )

Proof of Theorem feq123d

Step	Hyp	Ref	Expression
1		feq12d.1	. . 3 |- ( ph -> F = G )
2		feq12d.2	. . 3 |- ( ph -> A = B )
3	1, 2	feq12d 5397	. 2 |- ( ph -> ( F : A --> C <-> G : B --> C ) )
4		feq123d.3	. . 3 |- ( ph -> C = D )
5		feq3 5392	. . 3 |- ( C = D -> ( G : B --> C <-> G : B --> D ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( G : B --> C <-> G : B --> D ) )
7	3, 6	bitrd 188	1 |- ( ph -> ( F : A --> C <-> G : B --> D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   -->wf 5254

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262

This theorem is referenced by:  feq123  5399  feq23d  5403  csbwrdg  10949