Theorem spimt

Description: Closed theorem form of spim . (Contributed by NM, 15-Jan-2008) (Revised by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 21-Mar-2023) Usage of this theorem is discouraged because it depends on ax-13 . (New usage is discouraged.)

	Ref	Expression
Assertion	spimt	$$⊢\left(Ⅎx\psi \wedge \forall x\left(x=y\to \left(\phi \to \psi \right)\right)\right)\to \left(\forall x\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax6e	$$⊢\exists xx=y$$
2		exim	$$⊢\forall x\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\exists xx=y\to \exists x\left(\phi \to \psi \right)\right)$$
3	1 2	mpi	$$⊢\forall x\left(x=y\to \left(\phi \to \psi \right)\right)\to \exists x\left(\phi \to \psi \right)$$
4		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
5	3 4	sylib	$$⊢\forall x\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\forall x\phi \to \exists x\psi \right)$$
6		id	$$⊢Ⅎx\psi \to Ⅎx\psi$$
7	6	19.9d	$$⊢Ⅎx\psi \to \left(\exists x\psi \to \psi \right)$$
8	5 7	sylan9r	$$⊢\left(Ⅎx\psi \wedge \forall x\left(x=y\to \left(\phi \to \psi \right)\right)\right)\to \left(\forall x\phi \to \psi \right)$$