MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax67to7 Unicode version

Theorem ax67to7 2107
Description: Re-derivation of ax-7 1708 from ax67 2104. Note that ax-6o 2076 and ax-7 1708 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax67to7  |-  ( A. x A. y ph  ->  A. y A. x ph )

Proof of Theorem ax67to7
StepHypRef Expression
1 ax67to6 2106 . . 3  |-  ( -. 
A. y  -.  A. y  -.  A. x A. y ph  ->  -.  A. x A. y ph )
21con4i 122 . 2  |-  ( A. x A. y ph  ->  A. y  -.  A. y  -.  A. x A. y ph )
3 ax67 2104 . . 3  |-  ( -. 
A. y  -.  A. x A. y ph  ->  A. x ph )
43alimi 1546 . 2  |-  ( A. y  -.  A. y  -. 
A. x A. y ph  ->  A. y A. x ph )
52, 4syl 15 1  |-  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-7 1708  ax-4 2074  ax-5o 2075  ax-6o 2076
  Copyright terms: Public domain W3C validator