Theorem equid1 32478

Description: Proof of equid 1855 from our older axioms. 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 (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1758; see the proof of equid 1855. See equid1ALT 32496 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equid1	|- x = x

Proof of Theorem equid1

Step	Hyp	Ref	Expression
1		ax-c4 32456	. . . 4 |- ( A. x ( A. x -. A. x x = x -> ( x = x -> A. x x = x ) ) -> ( A. x -. A. x x = x -> A. x ( x = x -> A. x x = x ) ) )
2		ax-c5 32455	. . . . 5 |- ( A. x -. A. x x = x -> -. A. x x = x )
3		ax-c9 32462	. . . . 5 |- ( -. A. x x = x -> ( -. A. x x = x -> ( x = x -> A. x x = x ) ) )
4	2, 2, 3	sylc 62	. . . 4 |- ( A. x -. A. x x = x -> ( x = x -> A. x x = x ) )
5	1, 4	mpg 1671	. . 3 |- ( A. x -. A. x x = x -> A. x ( x = x -> A. x x = x ) )
6		ax-c10 32458	. . 3 |- ( A. x ( x = x -> A. x x = x ) -> x = x )
7	5, 6	syl 17	. 2 |- ( A. x -. A. x x = x -> x = x )
8		ax-c7 32457	. 2 |- ( -. A. x -. A. x x = x -> x = x )
9	7, 8	pm2.61i 168	1 |- x = x

Syntax hints:   -. wn 3   -> wi 4   A.wal 1442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-c5 32455  ax-c4 32456  ax-c7 32457  ax-c10 32458  ax-c9 32462

This theorem is referenced by: (None)