Theorem bj-nnf-spime 37040

Description: An existential generalization result in deduction form, 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-nnf-spime.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-nnf-spime.nf	⊢ (𝜑 → Ⅎ'𝑥𝜓)
bj-nnf-spime.is	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))

Assertion

Ref	Expression
bj-nnf-spime	⊢ (𝜑 → (𝜓 → ∃𝑥𝜒))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-nnf-spime

Step	Hyp	Ref	Expression
1		bj-nnf-spime.nf	. 2 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
2		ax6ev 1971	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		bj-nnf-spime.nf0	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4		bj-nnf-spime.is	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))
5	4	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	3, 5	eximdh 1866	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜓 → 𝜒)))
7	2, 6	mpi 20	. 2 ⊢ (𝜑 → ∃𝑥(𝜓 → 𝜒))
8		bj-19.37im 37029	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜓 → 𝜒) → (𝜓 → ∃𝑥𝜒)))
9	1, 7, 8	sylc 65	1 ⊢ (𝜑 → (𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  ∃wex 1781  Ⅎ'wnnf 36991

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  df-bj-nnf 36992

This theorem is referenced by: (None)