Theorem wl-lem-exsb

Description: This theorem provides a basic working step in proving theorems about E* or E! . (Contributed by Wolf Lammen, 3-Oct-2019)

	Ref	Expression
Assertion	wl-lem-exsb	$$⊢x=y\to \left(\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax12v2	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
2		sp	$$⊢\forall x\left(x=y\to \phi \right)\to \left(x=y\to \phi \right)$$
3	2	com12	$$⊢x=y\to \left(\forall x\left(x=y\to \phi \right)\to \phi \right)$$
4	1 3	impbid	$$⊢x=y\to \left(\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$