Theorem equidqe 1585

Description: equid 1753 with some quantification and negation without using ax-4 1563 or ax-17 1579. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)

Assertion

Ref	Expression
equidqe	|- -. A. y -. x = x

Proof of Theorem equidqe

Step	Hyp	Ref	Expression
1		ax-9 1584	. 2 |- -. A. y -. y = x
2		ax-8 1557	. . . . 5 |- ( y = x -> ( y = x -> x = x ) )
3	2	pm2.43i 49	. . . 4 |- ( y = x -> x = x )
4	3	con3i 641	. . 3 |- ( -. x = x -> -. y = x )
5	4	alimi 1508	. 2 |- ( A. y -. x = x -> A. y -. y = x )
6	1, 5	mto 672	1 |- -. A. y -. x = x

Syntax hints:   -. wn 3   A.wal 1400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-8 1557  ax-i9 1583

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is referenced by:  ax4sp1  1586