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Theorem equidALT 1819
Description: Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid 1818 from older axioms ax-6o 1697 and ax-9o 1815. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equidALT  |-  x  =  x

Proof of Theorem equidALT
StepHypRef Expression
1 ax-12o 1664 . . . . 5  |-  ( -. 
A. x  x  =  x  ->  ( -.  A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
) )
21pm2.43i 45 . . . 4  |-  ( -. 
A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
)
32alimi 1546 . . 3  |-  ( A. x  -.  A. x  x  =  x  ->  A. x
( x  =  x  ->  A. x  x  =  x ) )
4 ax-9o 1815 . . 3  |-  ( A. x ( x  =  x  ->  A. x  x  =  x )  ->  x  =  x )
53, 4syl 17 . 2  |-  ( A. x  -.  A. x  x  =  x  ->  x  =  x )
6 ax-6o 1697 . 2  |-  ( -. 
A. x  -.  A. x  x  =  x  ->  x  =  x )
75, 6pm2.61i 158 1  |-  x  =  x
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-12o 1664  ax-6o 1697  ax-9o 1815
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