Theorem ax12b

Description: A bidirectional version of axc15 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	ax12b	$$⊢\left(\neg \forall xx=y\wedge x=y\right)\to \left(\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		axc15	$$⊢\neg \forall xx=y\to \left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
2	1	imp	$$⊢\left(\neg \forall xx=y\wedge x=y\right)\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
3		sp	$$⊢\forall x\left(x=y\to \phi \right)\to \left(x=y\to \phi \right)$$
4	3	com12	$$⊢x=y\to \left(\forall x\left(x=y\to \phi \right)\to \phi \right)$$
5	4	adantl	$$⊢\left(\neg \forall xx=y\wedge x=y\right)\to \left(\forall x\left(x=y\to \phi \right)\to \phi \right)$$
6	2 5	impbid	$$⊢\left(\neg \forall xx=y\wedge x=y\right)\to \left(\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$