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

Theorem equid1 32231
Description: Proof of equid 1839 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 1748; see the proof of equid 1839. See equid1ALT 32249 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 32209 . . . 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 32208 . . . . 5  |-  ( A. x  -.  A. x  x  =  x  ->  -.  A. x  x  =  x )
3 ax-c9 32215 . . . . 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 1667 . . 3  |-  ( A. x  -.  A. x  x  =  x  ->  A. x
( x  =  x  ->  A. x  x  =  x ) )
6 ax-c10 32211 . . 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 32210 . 2  |-  ( -. 
A. x  -.  A. x  x  =  x  ->  x  =  x )
97, 8pm2.61i 167 1  |-  x  =  x
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-c5 32208  ax-c4 32209  ax-c7 32210  ax-c10 32211  ax-c9 32215
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator