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Theorem equid 1570
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 1567 . 2 y y = x
2 ax-17 1401 . . 3 (x = xy x = x)
3 ax-8 1377 . . . 4 (y = x → (y = xx = x))
43pm2.43i 43 . . 3 (y = xx = x)
52, 4exlimih 1467 . 2 (y y = xx = x)
61, 5ax-mp 7 1 x = x
Colors of variables: wff set class
Syntax hints:  wex 1361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1317  ax-ie2 1363  ax-8 1377  ax-17 1401  ax-i9 1405
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  nfequid  1571  stdpc6  1572  equcomi  1573  equveli  1624  sbid  1639  ax16i  1721  exists1  1978  vjust  2534  vex  2536  reu6  2705  nfccdeq  2737  sbc8g  2746  dfnul3  3205  rab0  3224  int0  3581  ruv  4182  relop  4379  f1eqcocnv  5323  mpt2xopoveq  5743  mathbox  5925
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