MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feq23i Structured version   Visualization version   GIF version

Theorem feq23i 6710
Description: Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq23i.1 𝐴 = 𝐶
feq23i.2 𝐵 = 𝐷
Assertion
Ref Expression
feq23i (𝐹:𝐴𝐵𝐹:𝐶𝐷)

Proof of Theorem feq23i
StepHypRef Expression
1 feq23i.1 . 2 𝐴 = 𝐶
2 feq23i.2 . 2 𝐵 = 𝐷
3 feq23 6700 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
41, 2, 3mp2an 688 1 (𝐹:𝐴𝐵𝐹:𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  wf 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-fn 6545  df-f 6546
This theorem is referenced by:  ftpg  7155  hashf  14302  funcoppc  17829  cnextfval  23786  uhgr0  28600  lfgredgge2  28651  mbfmvolf  33563  eulerpartlemt  33668  ismgmOLD  37021  elghomOLD  37058  tendoset  39933  pwssplit4  42133  isomushgr  46792  lincdifsn  47192
  Copyright terms: Public domain W3C validator