Theorem wl-ax11-lem9

Description: The easy part when x coincides with y . (Contributed by Wolf Lammen, 30-Jun-2019)

	Ref	Expression
Assertion	wl-ax11-lem9	$$⊢\forall xx=y\to \left(\forall y\forall x\phi \leftrightarrow \forall x\forall y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biidd	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \phi \right)$$
2	1	dral1	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\phi \right)$$
3	2	aecoms	$$⊢\forall yy=x\to \left(\forall x\phi \leftrightarrow \forall y\phi \right)$$
4	3	dral1	$$⊢\forall yy=x\to \left(\forall y\forall x\phi \leftrightarrow \forall x\forall y\phi \right)$$
5	4	aecoms	$$⊢\forall xx=y\to \left(\forall y\forall x\phi \leftrightarrow \forall x\forall y\phi \right)$$