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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  1881  sbid  1896  ax11eq  2109  exists1  2207  vjust  2764  nfccdeq  2964  sbc8g  2973  rab0  3450  dfid3  4282  reusv5OLD  4516  reusv7OLD  4518  relop  4822  fv2  5454  fsplit  6157  ruv  7282  konigthlem  8158  alexsubALTlem3  17705  isppw2  20315  avril1  20796  mathbox  22982  foo3  22983  domep  23518  dffix2  23821  elfuns  23829  vecval3b  24819  mamulid  26825  elnev  27006  ipo0  27020  ifr0  27021  a12lem1  28297
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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