Theorem bj-spim0 37568

Description: A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1999, where the only DV condition is on 𝑥, 𝑦 and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)

Hypotheses

Ref	Expression
bj-spim0.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-spim0.nf	⊢ (𝜑 → (∃𝑥𝜒 → 𝜒))
bj-spim0.is	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))

Assertion

Ref	Expression
bj-spim0	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)

Proof of Theorem bj-spim0

Step	Hyp	Ref	Expression
1		bj-spim0.nf0	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-spim0.nf	. 2 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒))
3		ax6ev 2001	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
4	3	a1i 11	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝑦)
5		bj-spim0.is	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))
6	1, 2, 4, 5	bj-spim 37525	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-6 1999

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  bj-nnf-spim  37676