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Theorem equid 1818
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 1628; see the proof of equid1 1820. See equidALT 1819 for an alternate proof. (Contributed by NM, 30-Nov-2008.) (Proof modification is discouraged.)
Assertion
Ref Expression
equid  |-  x  =  x

Proof of Theorem equid
StepHypRef Expression
1 ax-9 1684 . . 3  |-  -.  A. x  -.  x  =  x
2 hbn1 1564 . . . 4  |-  ( -. 
A. x  x  =  x  ->  A. x  -.  A. x  x  =  x )
3 ax-12o 1664 . . . . . . 7  |-  ( -. 
A. x  x  =  x  ->  ( -.  A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
) )
43pm2.43i 45 . . . . . 6  |-  ( -. 
A. x  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
)
54con3d 127 . . . . 5  |-  ( -. 
A. x  x  =  x  ->  ( -.  A. x  x  =  x  ->  -.  x  =  x ) )
65pm2.43i 45 . . . 4  |-  ( -. 
A. x  x  =  x  ->  -.  x  =  x )
72, 6alrimih 1553 . . 3  |-  ( -. 
A. x  x  =  x  ->  A. x  -.  x  =  x
)
81, 7mt3 173 . 2  |-  A. x  x  =  x
98a4i 1699 1  |-  x  =  x
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  stdpc6  1821  equcomi-o  1823  equveli  1880  sbid  1895  ax11eq  2105  exists1  2202  vjust  2728  nfccdeq  2919  sbc8g  2928  rab0  3382  dfid3  4203  reusv5OLD  4435  reusv7OLD  4437  relop  4741  fv2  5373  fsplit  6075  ruv  7198  konigthlem  8070  alexsubALTlem3  17575  isppw2  20185  avril1  20666  mathbox  22852  foo3  22853  domep  23317  dffix2  23620  elfuns  23628  vecval3b  24618  mamulid  26624  elnev  26805  ipo0  26819  ifr0  26820  a12lem1  27819
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-12o 1664  ax-9 1684  ax-4 1692
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