Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax8dfeq Structured version   Visualization version   GIF version

Theorem ax8dfeq 33117
Description: A version of ax-8 2116 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
ax8dfeq 𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))

Proof of Theorem ax8dfeq
StepHypRef Expression
1 ax6e 2402 . 2 𝑧 𝑧 = 𝑤
2 ax8 2120 . . . 4 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
32equcoms 2027 . . 3 (𝑧 = 𝑤 → (𝑤𝑥𝑧𝑥))
4 ax8 2120 . . 3 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
53, 4imim12d 81 . 2 (𝑧 = 𝑤 → ((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦)))
61, 5eximii 1838 1 𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1781
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator