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

Theorem feq23i 6481
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 6471 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
41, 2, 3mp2an 691 1 (𝐹:𝐴𝐵𝐹:𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wf 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-in 3917  df-ss 3927  df-fn 6331  df-f 6332
This theorem is referenced by:  ftpg  6891  hashf  13682  funcoppc  17123  cnextfval  22645  uhgr0  26844  lfgredgge2  26895  mbfmvolf  31531  eulerpartlemt  31636  ismgmOLD  35166  elghomOLD  35203  tendoset  37933  pwssplit4  39828  isomushgr  44136  lincdifsn  44624
  Copyright terms: Public domain W3C validator