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Theorem equid 1589
 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 1586 . 2 𝑦 𝑦 = 𝑥
2 ax-17 1419 . . 3 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
3 ax-8 1395 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
43pm2.43i 43 . . 3 (𝑦 = 𝑥𝑥 = 𝑥)
52, 4exlimih 1484 . 2 (∃𝑦 𝑦 = 𝑥𝑥 = 𝑥)
61, 5ax-mp 7 1 𝑥 = 𝑥
 Colors of variables: wff set class Syntax hints:  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-17 1419  ax-i9 1423 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  nfequid  1590  stdpc6  1591  equcomi  1592  equveli  1642  sbid  1657  ax16i  1738  exists1  1996  vjust  2558  vex  2560  reu6  2730  nfccdeq  2762  sbc8g  2771  dfnul3  3227  rab0  3246  int0  3629  ruv  4274  relop  4486  f1eqcocnv  5431  mpt2xopoveq  5855
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