Theorem equid 1723

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.

Step	Hyp	Ref	Expression
1		a9e 1718	. 2 ⊢ ∃𝑦 𝑦 = 𝑥
2		ax-17 1548	. . 3 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
3		ax-8 1526	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
4	3	pm2.43i 49	. . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
5	2, 4	exlimih 1615	. 2 ⊢ (∃𝑦 𝑦 = 𝑥 → 𝑥 = 𝑥)
6	1, 5	ax-mp 5	1 ⊢ 𝑥 = 𝑥

Syntax hints:  ∃wex 1514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1471  ax-ie2 1516  ax-8 1526  ax-17 1548  ax-i9 1552

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfequid  1724  stdpc6  1725  equcomi  1726  equveli  1781  sbid  1796  ax16i  1880  exists1  2149  vjust  2772  vex  2774  reu6  2961  nfccdeq  2995  sbc8g  3005  dfnul3  3462  rab0  3488  int0  3898  ruv  4596  dcextest  4627  relop  4826  f1eqcocnv  5850  mpoxopoveq  6316  snexxph  7034