Theorem bj-spime 36889

Description: A lemma for existential generalization. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1971 will prove Hypothesis bj-spime.denote. (Contributed by BJ, 4-Apr-2026.)

Hypotheses

Ref	Expression
bj-spime.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-spime.nf	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
bj-spime.denote	⊢ (𝜑 → ∃𝑥𝜓)
bj-spime.maj	⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))

Assertion

Ref	Expression
bj-spime	⊢ (𝜑 → (𝜒 → ∃𝑥𝜃))

Proof of Theorem bj-spime

Step	Hyp	Ref	Expression
1		bj-spime.nf	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
2		bj-spime.denote	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
3		bj-spime.nf0	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4		bj-spime.maj	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
5	4	ex 412	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
6	3, 5	eximdh 1866	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜒 → 𝜃)))
7	2, 6	mpd 15	. 2 ⊢ (𝜑 → ∃𝑥(𝜒 → 𝜃))
8		bj-spimenfa 36887	. 2 ⊢ ((𝜒 → ∀𝑥𝜒) → (∃𝑥(𝜒 → 𝜃) → (𝜒 → ∃𝑥𝜃)))
9	1, 7, 8	sylc 65	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜃))

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

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

This theorem is referenced by: (None)