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

Theorem zfinf 7857
Description: Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfinf  |-  E. x
( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
Distinct variable group:    x, y, z

Proof of Theorem zfinf
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-inf 7856 . 2  |-  E. x
( y  e.  x  /\  A. w ( w  e.  x  ->  E. z
( w  e.  z  /\  z  e.  x
) ) )
2 elequ1 1759 . . . . . 6  |-  ( w  =  y  ->  (
w  e.  x  <->  y  e.  x ) )
3 elequ1 1759 . . . . . . . 8  |-  ( w  =  y  ->  (
w  e.  z  <->  y  e.  z ) )
43anbi1d 704 . . . . . . 7  |-  ( w  =  y  ->  (
( w  e.  z  /\  z  e.  x
)  <->  ( y  e.  z  /\  z  e.  x ) ) )
54exbidv 1680 . . . . . 6  |-  ( w  =  y  ->  ( E. z ( w  e.  z  /\  z  e.  x )  <->  E. z
( y  e.  z  /\  z  e.  x
) ) )
62, 5imbi12d 320 . . . . 5  |-  ( w  =  y  ->  (
( w  e.  x  ->  E. z ( w  e.  z  /\  z  e.  x ) )  <->  ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) ) )
76cbvalv 1971 . . . 4  |-  ( A. w ( w  e.  x  ->  E. z
( w  e.  z  /\  z  e.  x
) )  <->  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) )
87anbi2i 694 . . 3  |-  ( ( y  e.  x  /\  A. w ( w  e.  x  ->  E. z
( w  e.  z  /\  z  e.  x
) ) )  <->  ( y  e.  x  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) ) )
98exbii 1634 . 2  |-  ( E. x ( y  e.  x  /\  A. w
( w  e.  x  ->  E. z ( w  e.  z  /\  z  e.  x ) ) )  <->  E. x ( y  e.  x  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) ) )
101, 9mpbi 208 1  |-  E. x
( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367   E.wex 1586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-inf 7856
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590
This theorem is referenced by:  axinf2  7858  axinfndlem1  8784
  Copyright terms: Public domain W3C validator