Theorem 2ax6elem

Description: We can always find values matching x and y , as long as they are represented by distinct variables. This theorem merges two ax6e instances E. z z = x and E. w w = y into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Wolf Lammen, 27-Sep-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	2ax6elem	|- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) )

Proof

Step	Hyp	Ref	Expression
1		ax6e	|- E. z z = x
2		nfnae	|- F/ z -. A. w w = z
3		nfnae	|- F/ z -. A. w w = x
4	2 3	nfan	|- F/ z ( -. A. w w = z /\ -. A. w w = x )
5		nfeqf	|- ( ( -. A. w w = z /\ -. A. w w = x ) -> F/ w z = x )
6		pm3.21	|- ( w = y -> ( z = x -> ( z = x /\ w = y ) ) )
7	5 6	spimed	|- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( z = x -> E. w ( z = x /\ w = y ) ) )
8	4 7	eximd	|- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( E. z z = x -> E. z E. w ( z = x /\ w = y ) ) )
9	1 8	mpi	|- ( ( -. A. w w = z /\ -. A. w w = x ) -> E. z E. w ( z = x /\ w = y ) )
10	9	ex	|- ( -. A. w w = z -> ( -. A. w w = x -> E. z E. w ( z = x /\ w = y ) ) )
11		ax6e	|- E. z z = y
12		nfae	|- F/ z A. w w = x
13		equvini	|- ( z = y -> E. w ( z = w /\ w = y ) )
14		equtrr	|- ( w = x -> ( z = w -> z = x ) )
15	14	anim1d	|- ( w = x -> ( ( z = w /\ w = y ) -> ( z = x /\ w = y ) ) )
16	15	aleximi	|- ( A. w w = x -> ( E. w ( z = w /\ w = y ) -> E. w ( z = x /\ w = y ) ) )
17	13 16	syl5	|- ( A. w w = x -> ( z = y -> E. w ( z = x /\ w = y ) ) )
18	12 17	eximd	|- ( A. w w = x -> ( E. z z = y -> E. z E. w ( z = x /\ w = y ) ) )
19	11 18	mpi	|- ( A. w w = x -> E. z E. w ( z = x /\ w = y ) )
20	10 19	pm2.61d2	|- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) )