Theorem eu1

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of Margaris p. 110. (Contributed by NM, 20-Aug-1993) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 29-Oct-2018) Avoid ax-13 . (Revised by Wolf Lammen, 7-Feb-2023)

	Ref	Expression
Hypothesis	eu1.nf	|- F/ y ph
Assertion	eu1	|- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Proof

Step	Hyp	Ref	Expression
1		eu1.nf	|- F/ y ph
2		nfs1v	|- F/ x [ y / x ] ph
3	2	euf	|- ( E! y [ y / x ] ph <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
4	1	sb8euv	|- ( E! x ph <-> E! y [ y / x ] ph )
5	1	sb6rfv	|- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
6		equcom	|- ( x = y <-> y = x )
7	6	imbi2i	|- ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
8	7	albii	|- ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
9	5 8	anbi12ci	|- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
10		albiim	|- ( A. y ( [ y / x ] ph <-> y = x ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
11	9 10	bitr4i	|- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> A. y ( [ y / x ] ph <-> y = x ) )
12	11	exbii	|- ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
13	3 4 12	3bitr4i	|- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )