Theorem bj-spimvwt

Description: Closed form of spimvw . See also spimt . (Contributed by BJ, 8-Nov-2021)

	Ref	Expression
Assertion	bj-spimvwt	$$⊢\forall x\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\forall x\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		alequexv	$$⊢\forall x\left(x=y\to \left(\phi \to \psi \right)\right)\to \exists x\left(\phi \to \psi \right)$$
2		19.36v	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \psi \right)$$
3	1 2	sylib	$$⊢\forall x\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\forall x\phi \to \psi \right)$$