MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax5el Structured version   Unicode version

Theorem ax5el 2245
Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1671 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax5el  |-  ( x  e.  y  ->  A. z  x  e.  y )
Distinct variable groups:    x, z    y, z

Proof of Theorem ax5el
StepHypRef Expression
1 ax-c14 2200 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )
2 ax-c16 2201 . 2  |-  ( A. z  z  =  x  ->  ( x  e.  y  ->  A. z  x  e.  y ) )
3 ax-c16 2201 . 2  |-  ( A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y ) )
41, 2, 3pm2.61ii 165 1  |-  ( x  e.  y  ->  A. z  x  e.  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c14 2200  ax-c16 2201
This theorem is referenced by:  dveel2ALT  2247
  Copyright terms: Public domain W3C validator