Theorem bj-spimenfa

Description: An existential generalization result: if ph holds and implies ps for at least one value of x , and if furthermore x is A. -weakly nonfree in ph , then ps holds for at least one value of x . (Contributed by BJ, 3-Apr-2026) Proof should not use 19.35 . (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-spimenfa	$$⊢\left(\phi \to \forall x\phi \right)\to \left(\exists x\left(\phi \to \psi \right)\to \left(\phi \to \exists x\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-eximcom	$$⊢\exists x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \exists x\psi \right)$$
2		imim1	$$⊢\left(\phi \to \forall x\phi \right)\to \left(\left(\forall x\phi \to \exists x\psi \right)\to \left(\phi \to \exists x\psi \right)\right)$$
3	1 2	syl5	$$⊢\left(\phi \to \forall x\phi \right)\to \left(\exists x\left(\phi \to \psi \right)\to \left(\phi \to \exists x\psi \right)\right)$$