Theorem feq23d 5732

Description: Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)

Hypotheses

Ref	Expression
feq23d.1	|- ( ph -> A = C )
feq23d.2	|- ( ph -> B = D )

Assertion

Ref	Expression
feq23d	|- ( ph -> ( F : A --> B <-> F : C --> D ) )

Proof of Theorem feq23d

Step	Hyp	Ref	Expression
1		eqidd 2458	. 2 |- ( ph -> F = F )
2		feq23d.1	. 2 |- ( ph -> A = C )
3		feq23d.2	. 2 |- ( ph -> B = D )
4	1, 2, 3	feq123d 5727	1 |- ( ph -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1395   -->wf 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598

This theorem is referenced by:  nvof1o  6187  axdc4uz  12096  isacs  15068  isfunc  15280  funcres  15312  funcpropd  15316  estrcco  15526  funcestrcsetclem9  15544  fullestrcsetc  15547  fullsetcestrc  15562  1stfcl  15593  2ndfcl  15594  evlfcl  15618  curf1cl  15624  yonedalem3b  15675  intopsn  16009  mhmpropd  16099  pwssplit1  17832  evls1sca  18487  islindf  18974  rrxds  21951  isgrp2d  25364  isgrpda  25426  isrngod  25508  rngosn3  25555  acunirnmpt  27653  cnmbfm  28407  elmrsubrn  29077  mapfzcons  30853  diophrw  30897  refsum2cnlem1  31615  mgmhmpropd  32735  funcringcsetcOLD2lem9  32996  funcringcsetclem9OLD  33019  aacllem  33360  islfld  34930  tendofset  36627  tendoset  36628