Theorem axc14 2113

Description: Axiom ax-c14 2222 is redundant if we assume ax-5 1704. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2112 and ax-5 1704. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc14

Proof of Theorem axc14

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbn1 1838	. . . . 5
2		dveel2 2112	. . . . 5
3	1, 2	hbim1 1918	. . . 4
4		elequ1 1821	. . . . 5
5	4	imbi2d 316	. . . 4
6	3, 5	dvelim 2079	. . 3
7		nfa1 1897	. . . . 5
8	7	nfn 1901	. . . 4
9	8	19.21 1905	. . 3
10	6, 9	syl6ib 226	. 2
11	10	pm2.86d 99	1

Syntax hints:  -.wn 3  ->wi 4  A.wal 1393

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-8 1820  ax-9 1822  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