Theorem bj-spim0 36931

Description: A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1969, 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 1971	. . 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 36888	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-6 1969

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  bj-nnf-spim  37039