Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  equid1 Structured version   Visualization version   Unicode version

Theorem equid1 32478
Description: Proof of equid 1855 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1758; see the proof of equid 1855. See equid1ALT 32496 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equid1  |-  x  =  x

Proof of Theorem equid1
StepHypRef Expression
1 ax-c4 32456 . . . 4  |-  ( A. x ( A. x  -.  A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
)  ->  ( A. x  -.  A. x  x  =  x  ->  A. x
( x  =  x  ->  A. x  x  =  x ) ) )
2 ax-c5 32455 . . . . 5  |-  ( A. x  -.  A. x  x  =  x  ->  -.  A. x  x  =  x )
3 ax-c9 32462 . . . . 5  |-  ( -. 
A. x  x  =  x  ->  ( -.  A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
) )
42, 2, 3sylc 62 . . . 4  |-  ( A. x  -.  A. x  x  =  x  ->  (
x  =  x  ->  A. x  x  =  x ) )
51, 4mpg 1671 . . 3  |-  ( A. x  -.  A. x  x  =  x  ->  A. x
( x  =  x  ->  A. x  x  =  x ) )
6 ax-c10 32458 . . 3  |-  ( A. x ( x  =  x  ->  A. x  x  =  x )  ->  x  =  x )
75, 6syl 17 . 2  |-  ( A. x  -.  A. x  x  =  x  ->  x  =  x )
8 ax-c7 32457 . 2  |-  ( -. 
A. x  -.  A. x  x  =  x  ->  x  =  x )
97, 8pm2.61i 168 1  |-  x  =  x
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-c5 32455  ax-c4 32456  ax-c7 32457  ax-c10 32458  ax-c9 32462
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator