Theorem spimw

Description: Specialization. Lemma 8 of KalishMontague p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017) (Proof shortened by Wolf Lammen, 7-Aug-2017)

	Ref	Expression
Hypotheses	spimw.1	$$⊢\neg \psi \to \forall x\neg \psi$$
	spimw.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	spimw	$$⊢\forall x\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		spimw.1	$$⊢\neg \psi \to \forall x\neg \psi$$
2		spimw.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
3		ax6v	$$⊢\neg \forall x\neg x=y$$
4	1 2	spimfw	$$⊢\neg \forall x\neg x=y\to \left(\forall x\phi \to \psi \right)$$
5	3 4	ax-mp	$$⊢\forall x\phi \to \psi$$