Theorem equid1 1253

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 1113 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 1112	. 2 ⊢ ∃y y = x
2		ax-17 1190	. . 3 ⊢ (x = x → ∀y x = x)
3		ax-8 1101	. . . 4 ⊢ (y = x → (y = x → x = x))
4	3	pm2.43i 64	. . 3 ⊢ (y = x → x = x)
5	2, 4	19.23ai 1040	. 2 ⊢ (∃y y = x → x = x)
6	1, 5	ax-mp 7	1 ⊢ x = x

Syntax hints:  ∃wex 956   = wceq 1099

This theorem is referenced by:  a12study 1355

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-8 1101  ax-9 1102  ax-17 1190

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957

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