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Theorem equid 1571
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.

This proof is similar to Tarski's and makes use of a dummy variable y. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)

Assertion
Ref Expression
equid x = x

Proof of Theorem equid
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 a9e 1568 . 2 y y = x
2 ax-17 1400 . . 3 (x = xy x = x)
3 ax-8 1376 . . . 4 (y = x → (y = xx = x))
43pm2.43i 43 . . 3 (y = xx = x)
52, 4exlimih 1466 . 2 (y y = xx = x)
61, 5ax-mp 7 1 x = x
Colors of variables: wff set class
Syntax hints:  wex 1362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1318  ax-ie2 1364  ax-8 1376  ax-17 1400  ax-i9 1404
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  nfequid  1572  stdpc6  1573  equcomi  1574  equveli  1624  sbid  1639  ax16i  1720  exists1  1978  vjust  2534  vex  2536  reu6  2705  nfccdeq  2737  sbc8g  2746  dfnul3  3202  rab0  3221  int0  3601  ruv  4210  relop  4411  f1eqcocnv  5354  mpt2xopoveq  5775  mathbox  6681
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