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

Theorem dveel2ALT 2267
Description: Version of dveel2 2100 using ax-16 2220 instead of ax-17 1626. (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 ax17el 2265 . 2  |-  ( z  e.  w  ->  A. x  z  e.  w )
2 ax17el 2265 . 2  |-  ( z  e.  y  ->  A. w  z  e.  y )
3 elequ2 1730 . 2  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
41, 2, 3dvelimh 2067 1  |-  ( -. 
A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-15 2219  ax-16 2220
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator