Theorem feq12d 5469

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 5466	. 2 |- ( ph -> ( F : A --> C <-> G : A --> C ) )
3		feq12d.2	. . 3 |- ( ph -> A = B )
4	3	feq2d 5467	. 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 1395   -->wf 5320

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328

This theorem is referenced by:  feq123d  5470  smoeq  6451  lmbr2  14928  lmff  14963  limccl  15373  ellimc3apf  15374  uhgr0e  15923  incistruhgr  15931  upgr1edc  15962  iswlk  16120  bj-charfundcALT  16340