Theorem feq12d 5353

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 5350	. 2 |- ( ph -> ( F : A --> C <-> G : A --> C ) )
3		feq12d.2	. . 3 |- ( ph -> A = B )
4	3	feq2d 5351	. 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 1353   -->wf 5210

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-fun 5216  df-fn 5217  df-f 5218

This theorem is referenced by:  feq123d  5354  smoeq  6287  lmbr2  13576  lmff  13611  limccl  13990  ellimc3apf  13991  bj-charfundcALT  14412