Theorem feq12d 6033

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 6030	. 2 |- ( ph -> ( F : A --> C <-> G : A --> C ) )
3		feq12d.2	. . 3 |- ( ph -> A = B )
4	3	feq2d 6031	. 2 |- ( ph -> ( G : A --> C <-> G : B --> C ) )
5	2, 4	bitrd 268	1 |- ( ph -> ( F : A --> C <-> G : B --> C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  feq123d  6034  fprg  6422  smoeq  7447  oif  8435  1fv  12458  catcisolem  16756  hofcl  16899  dmdprd  18397  dpjf  18456  pjf2  20058  mat1dimmul  20282  lmbr2  21063  lmff  21105  dfac14  21421  lmmbr2  23057  lmcau  23111  perfdvf  23667  dvnfre  23715  dvle  23770  dvfsumle  23784  dvfsumge  23785  dvmptrecl  23787  uhgr0e  25966  uhgrstrrepe  25973  incistruhgr  25974  upgr1e  26008  1hevtxdg1  26402  umgr2v2e  26421  iswlk  26506  0wlkons1  26982  resf1o  29505  ismeas  30262  omsmeas  30385  breprexplema  30708  mbfresfi  33456  sdclem1  33539  dfac21  37636  fnlimfvre  39906  climrescn  39980  fourierdlem74  40397  fourierdlem103  40426  fourierdlem104  40427  sge0iunmpt  40635  ismea  40668  isome  40708  sssmf  40947  smflimlem3  40981  smflimlem4  40982  isupwlk  41717