Theorem 2ax6elem 2193

Description: We can always find values matching and , as long as they are represented by distinct variables. This theorem merges two ax6e 2002 instances and into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 33331. (Contributed by Wolf Lammen, 27-Sep-2018.)

Assertion

Ref	Expression
2ax6elem

Proof of Theorem 2ax6elem

Step	Hyp	Ref	Expression
1		ax6e 2002	. . . 4
2		nfnae 2058	. . . . . 6
3		nfnae 2058	. . . . . 6
4	2, 3	nfan 1928	. . . . 5
5		nfeqf 2045	. . . . . 6
6		pm3.21 448	. . . . . 6
7	5, 6	spimed 2007	. . . . 5
8	4, 7	eximd 1882	. . . 4
9	1, 8	mpi 17	. . 3
10	9	ex 434	. 2
11		ax6e 2002	. . 3
12		nfae 2056	. . . 4
13		equvini 2087	. . . . 5
14		equtrr 1797	. . . . . . 7
15	14	anim1d 564	. . . . . 6
16	15	aleximi 1653	. . . . 5
17	13, 16	syl5 32	. . . 4
18	12, 17	eximd 1882	. . 3
19	11, 18	mpi 17	. 2
20	10, 19	pm2.61d2 160	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612

This theorem is referenced by:  2ax6e  2194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617