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Theorem equid1 2110
Description: Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. 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 thus requires no dummy variables. A simpler proof, similar to Tarki's, is possible if we make use of ax-17 1606; see the proof of equid 1662. See equid1ALT 2128 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equid1  |-  x  =  x

Proof of Theorem equid1
StepHypRef Expression
1 ax-5o 2088 . . . 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-4 2087 . . . . 5  |-  ( A. x  -.  A. x  x  =  x  ->  -.  A. x  x  =  x )
3 ax-12o 2094 . . . . 5  |-  ( -. 
A. x  x  =  x  ->  ( -.  A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
) )
42, 2, 3sylc 56 . . . 4  |-  ( A. x  -.  A. x  x  =  x  ->  (
x  =  x  ->  A. x  x  =  x ) )
51, 4mpg 1538 . . 3  |-  ( A. x  -.  A. x  x  =  x  ->  A. x
( x  =  x  ->  A. x  x  =  x ) )
6 ax-9o 2090 . . 3  |-  ( A. x ( x  =  x  ->  A. x  x  =  x )  ->  x  =  x )
75, 6syl 15 . 2  |-  ( A. x  -.  A. x  x  =  x  ->  x  =  x )
8 ax-6o 2089 . 2  |-  ( -. 
A. x  -.  A. x  x  =  x  ->  x  =  x )
97, 8pm2.61i 156 1  |-  x  =  x
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-4 2087  ax-5o 2088  ax-6o 2089  ax-9o 2090  ax-12o 2094
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