Theorem 2mos

Description: Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005) (Proof shortened by Wolf Lammen, 21-May-2025)

	Ref	Expression
Hypothesis	2mos.1	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion	2mos	|- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )

Proof

Step	Hyp	Ref	Expression
1		2mos.1	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2		2mo	|- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) )
3	1	2sbievw	|- ( [ z / x ] [ w / y ] ph <-> ps )
4	3	anbi2i	|- ( ( ph /\ [ z / x ] [ w / y ] ph ) <-> ( ph /\ ps ) )
5	4	imbi1i	|- ( ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) <-> ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
6	5	2albii	|- ( A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) <-> A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
7	6	2albii	|- ( A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
8	2 7	bitri	|- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )