Theorem ax12wlem

Description: Lemma for weak version of ax-12 . Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w . (Contributed by NM, 10-Apr-2017)

	Ref	Expression
Hypothesis	ax12wlemw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ax12wlem	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax12wlemw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		ax-5	$$⊢\psi \to \forall x\psi$$
3	1 2	ax12i	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$