Theorem 2ax6elem 2179

Description: We can always find values matching x and y, as long as they are represented by distinct variables. This theorem merges two ax6e 1971 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 32429. (Contributed by Wolf Lammen, 27-Sep-2018.)

Assertion

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

Proof of Theorem 2ax6elem

Step	Hyp	Ref	Expression
1		ax6e 1971	. . . 4 |- E. z z = x
2		nfnae 2031	. . . . . 6 |- F/ z -. A. w w = z
3		nfnae 2031	. . . . . 6 |- F/ z -. A. w w = x
4	2, 3	nfan 1875	. . . . 5 |- F/ z ( -. A. w w = z /\ -. A. w w = x )
5		nfeqf 2018	. . . . . 6 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> F/ w z = x )
6		nfr 1821	. . . . . . 7 |- ( F/ w z = x -> ( z = x -> A. w z = x ) )
7		ax6e 1971	. . . . . . . . 9 |- E. w w = y
8		pm3.21 448	. . . . . . . . 9 |- ( w = y -> ( z = x -> ( z = x /\ w = y ) ) )
9	7, 8	eximii 1637	. . . . . . . 8 |- E. w ( z = x -> ( z = x /\ w = y ) )
10	9	19.35i 1666	. . . . . . 7 |- ( A. w z = x -> E. w ( z = x /\ w = y ) )
11	6, 10	syl6 33	. . . . . 6 |- ( F/ w z = x -> ( z = x -> E. w ( z = x /\ w = y ) ) )
12	5, 11	syl 16	. . . . 5 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( z = x -> E. w ( z = x /\ w = y ) ) )
13	4, 12	eximd 1830	. . . 4 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( E. z z = x -> E. z E. w ( z = x /\ w = y ) ) )
14	1, 13	mpi 17	. . 3 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> E. z E. w ( z = x /\ w = y ) )
15	14	ex 434	. 2 |- ( -. A. w w = z -> ( -. A. w w = x -> E. z E. w ( z = x /\ w = y ) ) )
16		ax6e 1971	. . 3 |- E. z z = y
17		nfae 2029	. . . 4 |- F/ z A. w w = x
18		equvini 2060	. . . . 5 |- ( z = y -> E. w ( z = w /\ w = y ) )
19		equtrr 1746	. . . . . . 7 |- ( w = x -> ( z = w -> z = x ) )
20	19	anim1d 564	. . . . . 6 |- ( w = x -> ( ( z = w /\ w = y ) -> ( z = x /\ w = y ) ) )
21	20	aleximi 1632	. . . . 5 |- ( A. w w = x -> ( E. w ( z = w /\ w = y ) -> E. w ( z = x /\ w = y ) ) )
22	18, 21	syl5 32	. . . 4 |- ( A. w w = x -> ( z = y -> E. w ( z = x /\ w = y ) ) )
23	17, 22	eximd 1830	. . 3 |- ( A. w w = x -> ( E. z z = y -> E. z E. w ( z = x /\ w = y ) ) )
24	16, 23	mpi 17	. 2 |- ( A. w w = x -> E. z E. w ( z = x /\ w = y ) )
25	15, 24	pm2.61d2 160	1 |- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   A.wal 1377   E.wex 1596   F/wnf 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600

This theorem is referenced by:  2ax6e  2180