Theorem axc16g-o

Description: A generalization of axiom ax-c16 . Version of axc16g using ax-c11 . (Contributed by NM, 15-May-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	axc16g-o	$$⊢\forall xx=y\to \left(\phi \to \forall z\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		aev-o	$$⊢\forall xx=y\to \forall zz=x$$
2		ax-c16	$$⊢\forall xx=y\to \left(\phi \to \forall x\phi \right)$$
3		biidd	$$⊢\forall zz=x\to \left(\phi \leftrightarrow \phi \right)$$
4	3	dral1-o	$$⊢\forall zz=x\to \left(\forall z\phi \leftrightarrow \forall x\phi \right)$$
5	4	biimprd	$$⊢\forall zz=x\to \left(\forall x\phi \to \forall z\phi \right)$$
6	1 2 5	sylsyld	$$⊢\forall xx=y\to \left(\phi \to \forall z\phi \right)$$