Theorem feq12d 5518

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 5515	. 2 |- ( ph -> ( F : A --> C <-> G : A --> C ) )
3		feq12d.2	. . 3 |- ( ph -> A = B )
4	3	feq2d 5516	. 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 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:  feq123d  5519  smoeq  6551  lmbr2  15238  lmff  15273  limccl  15683  ellimc3apf  15684  uhgr0e  16237  incistruhgr  16245  upgr1edc  16276  umgr1een  16280  iswlk  16478  bj-charfundcALT  16749