MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax10lem4 Unicode version

Theorem ax10lem4 1986
Description: Lemma for ax10 1987. Similar to ax10o 1996 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.) (New usage discouraged.)
Assertion
Ref Expression
ax10lem4  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)

Proof of Theorem ax10lem4
StepHypRef Expression
1 ax-11 1753 . . 3  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
21sps 1762 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
3 pm2.27 37 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
43al2imi 1567 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
52, 4syld 42 1  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546
This theorem is referenced by:  ax10  1987  ax10OLD  1993  ax10NEW7  28813  dvelimhvAUX7  28832  dral1NEW7  28833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-ex 1548
  Copyright terms: Public domain W3C validator