Theorem feq12d 5479

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 )

Assertion

Ref	Expression
feq12d	|- ( ph -> ( F : A --> C <-> G : B --> C ) )

Proof of Theorem feq12d

Step	Hyp	Ref	Expression
1		feq12d.1	. . 3 |- ( ph -> F = G )
2	1	feq1d 5476	. 2 |- ( ph -> ( F : A --> C <-> G : A --> C ) )
3		feq12d.2	. . 3 |- ( ph -> A = B )
4	3	feq2d 5477	. 2 |- ( ph -> ( G : A --> C <-> G : B --> C ) )
5	2, 4	bitrd 188	1 |- ( ph -> ( F : A --> C <-> G : B --> C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  feq123d  5480  smoeq  6499  lmbr2  15008  lmff  15043  limccl  15453  ellimc3apf  15454  uhgr0e  16006  incistruhgr  16014  upgr1edc  16045  umgr1een  16049  iswlk  16247  bj-charfundcALT  16508