Theorem bj-spim0

Description: A universal specialization result in deduction form, proved 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-spim0.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-spim0.nf	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → 𝜒 ) )
	bj-spim0.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
Assertion	bj-spim0	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-spim0.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-spim0.nf	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → 𝜒 ) )
3		bj-spim0.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
4		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
5	4	a1i	⊢ ( 𝜑 → ∃ 𝑥 𝑥 = 𝑦 )
6	1 2 5 3	bj-spim	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )