Theorem bj-nnf-spime

Description: An existential generalization result in deduction form, from ax-1 -- ax-6 , where the only DV condition is on x , y , and where x should be nonfree in the new proposition ch (and in the context ph ). (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-spime.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-nnf-spime.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜓 )
	bj-nnf-spime.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
Assertion	bj-nnf-spime	⊢ ( 𝜑 → ( 𝜓 → ∃ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-spime.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-nnf-spime.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜓 )
3		bj-nnf-spime.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
4		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
5	3	ex	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
6	1 5	eximdh	⊢ ( 𝜑 → ( ∃ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 ( 𝜓 → 𝜒 ) ) )
7	4 6	mpi	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 → 𝜒 ) )
8		bj-19.37im	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∃ 𝑥 ( 𝜓 → 𝜒 ) → ( 𝜓 → ∃ 𝑥 𝜒 ) ) )
9	2 7 8	sylc	⊢ ( 𝜑 → ( 𝜓 → ∃ 𝑥 𝜒 ) )