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	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-spime.nf	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
	bj-spime.denote	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
	bj-spime.maj	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
Assertion	bj-spime	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-spime.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-spime.nf	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
3		bj-spime.denote	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
4		bj-spime.maj	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
5	4	ex	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
6	1 5	eximdh	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 ( 𝜒 → 𝜃 ) ) )
7	3 6	mpd	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜒 → 𝜃 ) )
8		bj-spimenfa	⊢ ( ( 𝜒 → ∀ 𝑥 𝜒 ) → ( ∃ 𝑥 ( 𝜒 → 𝜃 ) → ( 𝜒 → ∃ 𝑥 𝜃 ) ) )
9	2 7 8	sylc	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 𝜃 ) )