Theorem feq123d 5394

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 5393	. 2 |- ( ph -> ( F : A --> C <-> G : B --> C ) )
4		feq123d.3	. . 3 |- ( ph -> C = D )
5		feq3 5388	. . 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 5250

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  feq123  5395  feq23d  5399  csbwrdg  10943