ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  feq123d GIF version

Theorem feq123d 5117
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 5116 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
4 feq123d.3 . . 3 (𝜑𝐶 = 𝐷)
5 feq3 5112 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵𝐶𝐺:𝐵𝐷))
64, 5syl 14 . 2 (𝜑 → (𝐺:𝐵𝐶𝐺:𝐵𝐷))
73, 6bitrd 186 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wf 4977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-fun 4983  df-fn 4984  df-f 4985
This theorem is referenced by:  feq123  5118  feq23d  5122
  Copyright terms: Public domain W3C validator