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 1988 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 33199. (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 1988	. . . 4 |- E. z z = x
2		nfnae 2044	. . . . . 6 |- F/ z -. A. w w = z
3		nfnae 2044	. . . . . 6 |- F/ z -. A. w w = x
4	2, 3	nfan 1914	. . . . 5 |- F/ z ( -. A. w w = z /\ -. A. w w = x )
5		nfeqf 2031	. . . . . 6 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> F/ w z = x )
6		pm3.21 448	. . . . . 6 |- ( w = y -> ( z = x -> ( z = x /\ w = y ) ) )
7	5, 6	spimed 1993	. . . . 5 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> ( z = x -> E. w ( z = x /\ w = y ) ) )
8	4, 7	eximd 1868	. . . 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 17	. . 3 |- ( ( -. A. w w = z /\ -. A. w w = x ) -> E. z E. w ( z = x /\ w = y ) )
10	9	ex 434	. 2 |- ( -. A. w w = z -> ( -. A. w w = x -> E. z E. w ( z = x /\ w = y ) ) )
11		ax6e 1988	. . 3 |- E. z z = y
12		nfae 2042	. . . 4 |- F/ z A. w w = x
13		equvini 2073	. . . . 5 |- ( z = y -> E. w ( z = w /\ w = y ) )
14		equtrr 1783	. . . . . . 7 |- ( w = x -> ( z = w -> z = x ) )
15	14	anim1d 564	. . . . . 6 |- ( w = x -> ( ( z = w /\ w = y ) -> ( z = x /\ w = y ) ) )
16	15	aleximi 1640	. . . . 5 |- ( A. w w = x -> ( E. w ( z = w /\ w = y ) -> E. w ( z = x /\ w = y ) ) )
17	13, 16	syl5 32	. . . 4 |- ( A. w w = x -> ( z = y -> E. w ( z = x /\ w = y ) ) )
18	12, 17	eximd 1868	. . 3 |- ( A. w w = x -> ( E. z z = y -> E. z E. w ( z = x /\ w = y ) ) )
19	11, 18	mpi 17	. 2 |- ( A. w w = x -> E. z E. w ( z = x /\ w = y ) )
20	10, 19	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 1381   E.wex 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-nf 1604

This theorem is referenced by:  2ax6e  2180