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

Theorem hbae 2111
Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
Assertion
Ref Expression
hbae  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )

Proof of Theorem hbae
StepHypRef Expression
1 sp 1911 . . . . 5  |-  ( A. x  x  =  y  ->  x  =  y )
2 axc9 2102 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
31, 2syl7 71 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y ) ) )
4 axc112 1994 . . . 4  |-  ( A. z  z  =  x  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
5 axc11 2110 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
65pm2.43i 50 . . . . 5  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
7 axc112 1994 . . . . 5  |-  ( A. z  z  =  y  ->  ( A. y  x  =  y  ->  A. z  x  =  y )
)
86, 7syl5 34 . . . 4  |-  ( A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
93, 4, 8pm2.61ii 169 . . 3  |-  ( A. x  x  =  y  ->  A. z  x  =  y )
109axc4i 1954 . 2  |-  ( A. x  x  =  y  ->  A. x A. z  x  =  y )
11 ax-11 1893 . 2  |-  ( A. x A. z  x  =  y  ->  A. z A. x  x  =  y )
1210, 11syl 17 1  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-nf 1665
This theorem is referenced by:  nfae  2112  hbnae  2113  aevALT  2119  drex2  2126  ax6e2eq  36788
  Copyright terms: Public domain W3C validator