Theorem bj-spime

Description: A lemma for existential generalization. In applications, x = y will be substituted for ps and ax6ev will prove Hypothesis bj-spime.denote. (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-spime.nf0	$$⊢\phi \to \forall x\phi$$
	bj-spime.nf	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
	bj-spime.denote	$$⊢\phi \to \exists x\psi$$
	bj-spime.maj	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \to \theta \right)$$
Assertion	bj-spime	$$⊢\phi \to \left(\chi \to \exists x\theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-spime.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-spime.nf	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
3		bj-spime.denote	$$⊢\phi \to \exists x\psi$$
4		bj-spime.maj	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \to \theta \right)$$
5	4	ex	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
6	1 5	eximdh	$$⊢\phi \to \left(\exists x\psi \to \exists x\left(\chi \to \theta \right)\right)$$
7	3 6	mpd	$$⊢\phi \to \exists x\left(\chi \to \theta \right)$$
8		bj-spimenfa	$$⊢\left(\chi \to \forall x\chi \right)\to \left(\exists x\left(\chi \to \theta \right)\to \left(\chi \to \exists x\theta \right)\right)$$
9	2 7 8	sylc	$$⊢\phi \to \left(\chi \to \exists x\theta \right)$$