MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax11 Unicode version

Theorem ax11 2096
Description: Rederivation of axiom ax-11 1716 from ax-11o 2083, ax-10o 2081, and other older axioms. The proof does not require ax-16 2086 or ax-17 1604. See theorem ax11o 1937 for the derivation of ax-11o 2083 from ax-11 1716.

An open problem is whether we can prove this using ax-10 2082 instead of ax-10o 2081.

This proof uses newer axioms ax-5 1545 and ax-9 1637, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2078 and ax-9o 2080. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax11  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )

Proof of Theorem ax11
StepHypRef Expression
1 biidd 230 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1-o 2095 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 7 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1547 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 222 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 24 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 ax-4 2077 . . 3  |-  ( A. y ph  ->  ph )
8 ax-11o 2083 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
97, 8syl7 65 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
106, 9pm2.61i 158 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1528
This theorem is referenced by:  ax10o-o  2143
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-7 1709  ax-4 2077  ax-5o 2078  ax-6o 2079  ax-10o 2081  ax-11o 2083  ax-12o 2084
This theorem depends on definitions:  df-bi 179
  Copyright terms: Public domain W3C validator