Theorem axc16i

Description: Inference with axc16 as its conclusion. (Contributed by NM, 20-May-2008) (Proof modification is discouraged.) Usage of this theorem is discouraged because it depends on ax-13 . Use axc16 instead. (New usage is discouraged.)

	Ref	Expression
Hypotheses	axc16i.1	$$⊢x=z\to \left(\phi \leftrightarrow \psi \right)$$
	axc16i.2	$$⊢\psi \to \forall x\psi$$
Assertion	axc16i	$$⊢\forall xx=y\to \left(\phi \to \forall x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		axc16i.1	$$⊢x=z\to \left(\phi \leftrightarrow \psi \right)$$
2		axc16i.2	$$⊢\psi \to \forall x\psi$$
3		nfv	$$⊢Ⅎzx=y$$
4		nfv	$$⊢Ⅎxz=y$$
5		ax7	$$⊢x=z\to \left(x=y\to z=y\right)$$
6	3 4 5	cbv3	$$⊢\forall xx=y\to \forall zz=y$$
7		ax7	$$⊢z=x\to \left(z=y\to x=y\right)$$
8	7	spimvw	$$⊢\forall zz=y\to x=y$$
9		equcomi	$$⊢x=y\to y=x$$
10		equcomi	$$⊢z=y\to y=z$$
11		ax7	$$⊢y=z\to \left(y=x\to z=x\right)$$
12	10 11	syl	$$⊢z=y\to \left(y=x\to z=x\right)$$
13	9 12	syl5com	$$⊢x=y\to \left(z=y\to z=x\right)$$
14	13	alimdv	$$⊢x=y\to \left(\forall zz=y\to \forall zz=x\right)$$
15	8 14	mpcom	$$⊢\forall zz=y\to \forall zz=x$$
16		equcomi	$$⊢z=x\to x=z$$
17	16	alimi	$$⊢\forall zz=x\to \forall zx=z$$
18	15 17	syl	$$⊢\forall zz=y\to \forall zx=z$$
19	1	biimpcd	$$⊢\phi \to \left(x=z\to \psi \right)$$
20	19	alimdv	$$⊢\phi \to \left(\forall zx=z\to \forall z\psi \right)$$
21	2	nf5i	$$⊢Ⅎx\psi$$
22		nfv	$$⊢Ⅎz\phi$$
23	1	biimprd	$$⊢x=z\to \left(\psi \to \phi \right)$$
24	16 23	syl	$$⊢z=x\to \left(\psi \to \phi \right)$$
25	21 22 24	cbv3	$$⊢\forall z\psi \to \forall x\phi$$
26	20 25	syl6com	$$⊢\forall zx=z\to \left(\phi \to \forall x\phi \right)$$
27	6 18 26	3syl	$$⊢\forall xx=y\to \left(\phi \to \forall x\phi \right)$$