Theorem axc11nALT 2310

Description: Alternate proof of axc11n 2307 from axc11nlemALT 2306. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11nALT	|- ( A. x x = y -> A. y y = x )

Proof of Theorem axc11nALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcomi 1944	. . . . 5 |- ( z = x -> x = z )
2		dveeq1 2300	. . . . 5 |- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
3	1, 2	syl5com 31	. . . 4 |- ( z = x -> ( -. A. y y = x -> A. y x = z ) )
4		axc11r 2187	. . . . 5 |- ( A. x x = y -> ( A. y x = z -> A. x x = z ) )
5		axc11nlemALT 2306	. . . . 5 |- ( A. x x = z -> A. y y = x )
6	4, 5	syl6 35	. . . 4 |- ( A. x x = y -> ( A. y x = z -> A. y y = x ) )
7	3, 6	syl9 77	. . 3 |- ( z = x -> ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) ) )
8		ax6ev 1890	. . 3 |- E. z z = x
9	7, 8	exlimiiv 1859	. 2 |- ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) )
10	9	pm2.18d 124	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)