MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dveel2ALT Structured version   Unicode version

Theorem dveel2ALT 2271
Description: Alternate proof of dveel2 2114 using ax-c16 2225 instead of ax-5 1709. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveel2ALT  |-  ( -. 
A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
)
Distinct variable group:    x, z

Proof of Theorem dveel2ALT
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax5el 2269 . 2  |-  ( z  e.  w  ->  A. x  z  e.  w )
2 ax5el 2269 . 2  |-  ( z  e.  y  ->  A. w  z  e.  y )
3 elequ2 1828 . 2  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
41, 2, 3dvelimh 2082 1  |-  ( -. 
A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-c14 2224  ax-c16 2225
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator