Theorem feq123d 5322

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 5321	. 2 |- ( ph -> ( F : A --> C <-> G : B --> C ) )
4		feq123d.3	. . 3 |- ( ph -> C = D )
5		feq3 5316	. . 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 187	1 |- ( ph -> ( F : A --> C <-> G : B --> D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1342   -->wf 5178

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-fun 5184  df-fn 5185  df-f 5186

This theorem is referenced by:  feq123  5323  feq23d  5327