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Theorem feq123d 6500
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 6499 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
4 feq123d.3 . . 3 (𝜑𝐶 = 𝐷)
54feq3d 6498 . 2 (𝜑 → (𝐺:𝐵𝐶𝐺:𝐵𝐷))
63, 5bitrd 280 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wf 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-fun 6354  df-fn 6355  df-f 6356
This theorem is referenced by:  feq123  6501  feq23d  6506  fprg  6913  csbwrdg  13885  funcestrcsetclem8  17387  funcsetcestrclem8  17402  funcsetcestrclem9  17403  evlfcl  17462  yonedalem3a  17514  yonedalem4c  17517  yonedalem3b  17519  yonedainv  17521  iscau  23794  isuhgr  26759  uhgreq12g  26764  isuhgrop  26769  uhgrun  26773  isupgr  26783  upgrop  26793  isumgr  26794  upgrun  26817  umgrun  26819  lfuhgr1v0e  26950  wlkp1  27377  sseqf  31536  ismfs  32680  isrngo  35043  gneispace2  40347  funcringcsetcALTV2lem8  44146  funcringcsetclem8ALTV  44169
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