Theorem euf

Description: Version of eu6 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993) (Proof shortened by Wolf Lammen, 30-Oct-2018) Avoid ax-13 . (Revised by Wolf Lammen, 16-Oct-2022)

	Ref	Expression
Hypothesis	euf.1	|- F/ y ph
Assertion	euf	|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

Proof

Step	Hyp	Ref	Expression
1		euf.1	|- F/ y ph
2		eu6	|- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
3		nfv	|- F/ y x = z
4	1 3	nfbi	|- F/ y ( ph <-> x = z )
5	4	nfal	|- F/ y A. x ( ph <-> x = z )
6		nfv	|- F/ z A. x ( ph <-> x = y )
7		equequ2	|- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d	|- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv	|- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	5 6 9	cbvexv1	|- ( E. z A. x ( ph <-> x = z ) <-> E. y A. x ( ph <-> x = y ) )
11	2 10	bitri	|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )