Theorem equid1 2237

Description: Proof of equid 1791 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 1704; see the proof of equid 1791. See equid1ALT 2255 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equid1

Proof of Theorem equid1

Step	Hyp	Ref	Expression
1		ax-c4 2215	. . . 4
2		ax-c5 2214	. . . . 5
3		ax-c9 2221	. . . . 5
4	2, 2, 3	sylc 60	. . . 4
5	1, 4	mpg 1620	. . 3
6		ax-c10 2217	. . 3
7	5, 6	syl 16	. 2
8		ax-c7 2216	. 2
9	7, 8	pm2.61i 164	1

Syntax hints:  -.wn 3  ->wi 4  A.wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c10 2217  ax-c9 2221