Theorem axc14 2164

Description: Axiom ax-c14 32216 is redundant if we assume ax-5 1748. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that w is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2163 and ax-5 1748. 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	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x e. y -> A. z x e. y ) ) )

Proof of Theorem axc14

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbn1 1887	. . . . 5 |- ( -. A. z z = y -> A. z -. A. z z = y )
2		dveel2 2163	. . . . 5 |- ( -. A. z z = y -> ( w e. y -> A. z w e. y ) )
3	1, 2	hbim1 1973	. . . 4 |- ( ( -. A. z z = y -> w e. y ) -> A. z ( -. A. z z = y -> w e. y ) )
4		elequ1 1870	. . . . 5 |- ( w = x -> ( w e. y <-> x e. y ) )
5	4	imbi2d 317	. . . 4 |- ( w = x -> ( ( -. A. z z = y -> w e. y ) <-> ( -. A. z z = y -> x e. y ) ) )
6	3, 5	dvelim 2132	. . 3 |- ( -. A. z z = x -> ( ( -. A. z z = y -> x e. y ) -> A. z ( -. A. z z = y -> x e. y ) ) )
7		nfa1 1951	. . . . 5 |- F/ z A. z z = y
8	7	nfn 1955	. . . 4 |- F/ z -. A. z z = y
9	8	19.21 1959	. . 3 |- ( A. z ( -. A. z z = y -> x e. y ) <-> ( -. A. z z = y -> A. z x e. y ) )
10	6, 9	syl6ib 229	. 2 |- ( -. A. z z = x -> ( ( -. A. z z = y -> x e. y ) -> ( -. A. z z = y -> A. z x e. y ) ) )
11	10	pm2.86d 102	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x e. y -> A. z x e. y ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664

This theorem is referenced by: (None)