Theorem axc16i

Description: Inference with axc16 as its conclusion. Usage of axc16 is preferred since it requires fewer axioms. (Contributed by NM, 20-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axc16i.1	|- ( x = z -> ( ph <-> ps ) )
	axc16i.2	|- ( ps -> A. x ps )
Assertion	axc16i	|- ( A. x x = y -> ( ph -> A. x ph ) )

Proof

Step	Hyp	Ref	Expression
1		axc16i.1	|- ( x = z -> ( ph <-> ps ) )
2		axc16i.2	|- ( ps -> A. x ps )
3		nfv	|- F/ z x = y
4		nfv	|- F/ x z = y
5		ax7	|- ( x = z -> ( x = y -> z = y ) )
6	3 4 5	cbv3	|- ( A. x x = y -> A. z z = y )
7		ax7	|- ( z = x -> ( z = y -> x = y ) )
8	7	spimvw	|- ( A. z z = y -> x = y )
9		equcomi	|- ( x = y -> y = x )
10		equcomi	|- ( z = y -> y = z )
11		ax7	|- ( y = z -> ( y = x -> z = x ) )
12	10 11	syl	|- ( z = y -> ( y = x -> z = x ) )
13	9 12	syl5com	|- ( x = y -> ( z = y -> z = x ) )
14	13	alimdv	|- ( x = y -> ( A. z z = y -> A. z z = x ) )
15	8 14	mpcom	|- ( A. z z = y -> A. z z = x )
16		equcomi	|- ( z = x -> x = z )
17	16	alimi	|- ( A. z z = x -> A. z x = z )
18	15 17	syl	|- ( A. z z = y -> A. z x = z )
19	1	biimpcd	|- ( ph -> ( x = z -> ps ) )
20	19	alimdv	|- ( ph -> ( A. z x = z -> A. z ps ) )
21	2	nf5i	|- F/ x ps
22		nfv	|- F/ z ph
23	1	biimprd	|- ( x = z -> ( ps -> ph ) )
24	16 23	syl	|- ( z = x -> ( ps -> ph ) )
25	21 22 24	cbv3	|- ( A. z ps -> A. x ph )
26	20 25	syl6com	|- ( A. z x = z -> ( ph -> A. x ph ) )
27	6 18 26	3syl	|- ( A. x x = y -> ( ph -> A. x ph ) )