Theorem equid 1725

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 1720	. 2 |- E. y y = x
2		ax-17 1550	. . 3 |- ( x = x -> A. y x = x )
3		ax-8 1528	. . . 4 |- ( y = x -> ( y = x -> x = x ) )
4	3	pm2.43i 49	. . 3 |- ( y = x -> x = x )
5	2, 4	exlimih 1617	. 2 |- ( E. y y = x -> x = x )
6	1, 5	ax-mp 5	1 |- x = x

Syntax hints:   E.wex 1516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1473  ax-ie2 1518  ax-8 1528  ax-17 1550  ax-i9 1554

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfequid  1726  stdpc6  1727  equcomi  1728  equveli  1783  sbid  1798  ax16i  1882  exists1  2152  vjust  2777  vex  2779  reu6  2969  nfccdeq  3003  sbc8g  3013  dfnul3  3471  rab0  3497  int0  3913  ruv  4616  dcextest  4647  relop  4846  f1eqcocnv  5883  mpoxopoveq  6349  snexxph  7078