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Theorem feq123d 5947
 Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1 (𝜑𝐹 = 𝐺)
feq12d.2 (𝜑𝐴 = 𝐵)
feq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
feq123d (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))

Proof of Theorem feq123d
StepHypRef Expression
1 feq12d.1 . . 3 (𝜑𝐹 = 𝐺)
2 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
31, 2feq12d 5946 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
4 feq123d.3 . . 3 (𝜑𝐶 = 𝐷)
54feq3d 5945 . 2 (𝜑 → (𝐺:𝐵𝐶𝐺:𝐵𝐷))
63, 5bitrd 267 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  feq123  5948  feq23d  5953  fprg  6327  csbwrdg  13189  funcestrcsetclem8  16610  funcsetcestrclem8  16625  funcsetcestrclem9  16626  evlfcl  16685  yonedalem3a  16737  yonedalem4c  16740  yonedalem3b  16742  yonedainv  16744  iscau  22882  isuhgr  25726  uhgreq12g  25731  isuhgrop  25736  uhgrun  25740  isupgr  25751  isumgr  25761  upgrun  25784  umgrun  25786  isuhgra  25827  uhgraeq12d  25836  constr3trllem3  26180  sseqf  29781  ismfs  30700  isrngo  32866  gneispace2  37450  uhgruhgra  40375  uhgrauhgr  40376  lfuhgr1v0e  40480  1wlkp1  40890  funcringcsetcALTV2lem8  41835  funcringcsetclem8ALTV  41858
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