Theorem 2ax6elem 2278

Description: We can always find values matching x and y, as long as they are represented by distinct variables. This theorem merges two ax6e 2094 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 36925. (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 2094	. . . 4 |- E. z z = x
2		nfnae 2152	. . . . . 6 |- F/ z -. A. w w = z
3		nfnae 2152	. . . . . 6 |- F/ z -. A. w w = x
4	2, 3	nfan 2011	. . . . 5 |- F/ z ( -. A. w w = z /\ -. A. w w = x )
5		nfeqf 2139	. . . . . 6 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> F/ w z = x )
6		pm3.21 450	. . . . . 6 |- ( w = y -> ( z = x -> ( z = x /\ w = y ) ) )
7	5, 6	spimed 2099	. . . . 5 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( z = x -> E. w ( z = x /\ w = y ) ) )
8	4, 7	eximd 1960	. . . 4 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( E. z z = x -> E. z E. w ( z = x /\ w = y ) ) )
9	1, 8	mpi 20	. . 3 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> E. z E. w ( z = x /\ w = y ) )
10	9	ex 436	. 2 |- ( -. A. w w = z -> ( -. A. w w = x -> E. z E. w ( z = x /\ w = y ) ) )
11		ax6e 2094	. . 3 |- E. z z = y
12		nfae 2150	. . . 4 |- F/ z A. w w = x
13		equvini 2179	. . . . 5 |- ( z = y -> E. w ( z = w /\ w = y ) )
14		equtrr 1866	. . . . . . 7 |- ( w = x -> ( z = w -> z = x ) )
15	14	anim1d 568	. . . . . 6 |- ( w = x -> ( ( z = w /\ w = y ) -> ( z = x /\ w = y ) ) )
16	15	aleximi 1704	. . . . 5 |- ( A. w w = x -> ( E. w ( z = w /\ w = y ) -> E. w ( z = x /\ w = y ) ) )
17	13, 16	syl5 33	. . . 4 |- ( A. w w = x -> ( z = y -> E. w ( z = x /\ w = y ) ) )
18	12, 17	eximd 1960	. . 3 |- ( A. w w = x -> ( E. z z = y -> E. z E. w ( z = x /\ w = y ) ) )
19	11, 18	mpi 20	. 2 |- ( A. w w = x -> E. z E. w ( z = x /\ w = y ) )
20	10, 19	pm2.61d2 164	1 |- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   A.wal 1442   E.wex 1663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668

This theorem is referenced by:  2ax6e  2279