Theorem spimfw

Description: Specialization, with additional weakening (compared to sp ) to allow bundling of x and y . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017) (Proof shortened by Wolf Lammen, 7-Aug-2017)

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

Proof

Step	Hyp	Ref	Expression
1		spimfw.1	$$⊢\neg \psi \to \forall x\neg \psi$$
2		spimfw.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
3	2	speimfw	$$⊢\neg \forall x\neg x=y\to \left(\forall x\phi \to \exists x\psi \right)$$
4		df-ex	$$⊢\exists x\psi \leftrightarrow \neg \forall x\neg \psi$$
5	1	con1i	$$⊢\neg \forall x\neg \psi \to \psi$$
6	4 5	sylbi	$$⊢\exists x\psi \to \psi$$
7	3 6	syl6	$$⊢\neg \forall x\neg x=y\to \left(\forall x\phi \to \psi \right)$$