MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ax6elem Structured version   Unicode version

Theorem 2ax6elem 2179
Description: We can always find values matching  x and  y, as long as they are represented by distinct variables. This theorem merges two ax6e 1971 instances  E. z z  =  x and  E. w w  =  y into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 32429. (Contributed by Wolf Lammen, 27-Sep-2018.)
Assertion
Ref Expression
2ax6elem  |-  ( -. 
A. w  w  =  z  ->  E. z E. w ( z  =  x  /\  w  =  y ) )

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 1971 . . . 4  |-  E. z 
z  =  x
2 nfnae 2031 . . . . . 6  |-  F/ z  -.  A. w  w  =  z
3 nfnae 2031 . . . . . 6  |-  F/ z  -.  A. w  w  =  x
42, 3nfan 1875 . . . . 5  |-  F/ z ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )
5 nfeqf 2018 . . . . . 6  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  F/ w  z  =  x )
6 nfr 1821 . . . . . . 7  |-  ( F/ w  z  =  x  ->  ( z  =  x  ->  A. w  z  =  x )
)
7 ax6e 1971 . . . . . . . . 9  |-  E. w  w  =  y
8 pm3.21 448 . . . . . . . . 9  |-  ( w  =  y  ->  (
z  =  x  -> 
( z  =  x  /\  w  =  y ) ) )
97, 8eximii 1637 . . . . . . . 8  |-  E. w
( z  =  x  ->  ( z  =  x  /\  w  =  y ) )
10919.35i 1666 . . . . . . 7  |-  ( A. w  z  =  x  ->  E. w ( z  =  x  /\  w  =  y ) )
116, 10syl6 33 . . . . . 6  |-  ( F/ w  z  =  x  ->  ( z  =  x  ->  E. w
( z  =  x  /\  w  =  y ) ) )
125, 11syl 16 . . . . 5  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  ( z  =  x  ->  E. w
( z  =  x  /\  w  =  y ) ) )
134, 12eximd 1830 . . . 4  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  ( E. z  z  =  x  ->  E. z E. w
( z  =  x  /\  w  =  y ) ) )
141, 13mpi 17 . . 3  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  E. z E. w ( z  =  x  /\  w  =  y ) )
1514ex 434 . 2  |-  ( -. 
A. w  w  =  z  ->  ( -.  A. w  w  =  x  ->  E. z E. w
( z  =  x  /\  w  =  y ) ) )
16 ax6e 1971 . . 3  |-  E. z 
z  =  y
17 nfae 2029 . . . 4  |-  F/ z A. w  w  =  x
18 equvini 2060 . . . . 5  |-  ( z  =  y  ->  E. w
( z  =  w  /\  w  =  y ) )
19 equtrr 1746 . . . . . . 7  |-  ( w  =  x  ->  (
z  =  w  -> 
z  =  x ) )
2019anim1d 564 . . . . . 6  |-  ( w  =  x  ->  (
( z  =  w  /\  w  =  y )  ->  ( z  =  x  /\  w  =  y ) ) )
2120aleximi 1632 . . . . 5  |-  ( A. w  w  =  x  ->  ( E. w ( z  =  w  /\  w  =  y )  ->  E. w ( z  =  x  /\  w  =  y ) ) )
2218, 21syl5 32 . . . 4  |-  ( A. w  w  =  x  ->  ( z  =  y  ->  E. w ( z  =  x  /\  w  =  y ) ) )
2317, 22eximd 1830 . . 3  |-  ( A. w  w  =  x  ->  ( E. z  z  =  y  ->  E. z E. w ( z  =  x  /\  w  =  y ) ) )
2416, 23mpi 17 . 2  |-  ( A. w  w  =  x  ->  E. z E. w
( z  =  x  /\  w  =  y ) )
2515, 24pm2.61d2 160 1  |-  ( -. 
A. w  w  =  z  ->  E. z E. w ( z  =  x  /\  w  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596   F/wnf 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by:  2ax6e  2180
  Copyright terms: Public domain W3C validator