Theorem equid1 1484

Description: Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This proof is similar to Tarski's and makes use of a dummy variable y. See equid 1322 for a proof that avoids dummy variables (but is less intuitive).

Assertion

Ref	Expression
equid1	|- x = x

Proof of Theorem equid1

Step	Hyp	Ref	Expression
1		a9e 1321	. 2 |- E.y y = x
2		ax-17 1155	. . 3 |- (x = x -> A.y x = x)
3		ax-8 1144	. . . 4 |- (y = x -> (y = x -> x = x))
4	3	pm2.43i 78	. . 3 |- (y = x -> x = x)
5	2, 4	19.23ai 1250	. 2 |- (E.y y = x -> x = x)
6	1, 5	ax-mp 7	1 |- x = x

Syntax hints:   = wceq 1136  E.wex 1164

This theorem is referenced by:  ax16i 1485  a12study 1607

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1143  ax-8 1144  ax-9 1145  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162

This theorem depends on definitions:  df-bi 163  df-an 241  df-ex 1165

Copyright terms: Public domain