MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feq23 Structured version   Visualization version   GIF version

Theorem feq23 6190
Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
feq23 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))

Proof of Theorem feq23
StepHypRef Expression
1 feq2 6188 . 2 (𝐴 = 𝐶 → (𝐹:𝐴𝐵𝐹:𝐶𝐵))
2 feq3 6189 . 2 (𝐵 = 𝐷 → (𝐹:𝐶𝐵𝐹:𝐶𝐷))
31, 2sylan9bb 738 1 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729  df-fn 6052  df-f 6053
This theorem is referenced by:  feq23i  6200  ismgmOLD  33962  ismndo2  33986  rngomndo  34047  seff  39010
  Copyright terms: Public domain W3C validator