Theorem equidqe 2173

Description: equid 1676 with existential quantifier without using ax-4 2135 or ax-17 1616. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem equidqe

Step	Hyp	Ref	Expression
1		ax9from9o 2148	. 2 |- -. A.y -. y = x
2		ax-8 1675	. . . . 5 |- (y = x -> (y = x -> x = x))
3	2	pm2.43i 43	. . . 4 |- (y = x -> x = x)
4	3	con3i 127	. . 3 |- (-. x = x -> -. y = x)
5	4	alimi 1559	. 2 |- (A.y -. x = x -> A.y -. y = x)
6	1, 5	mto 167	1 |- -. A.y -. x = x

Syntax hints:  -. wn 3  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-8 1675  ax-6o 2137  ax-9o 2138

This theorem is referenced by:  ax4sp1  2174  equidq  2175