Theorem bj-spim 37173

Description: A lemma for universal specification. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1996 will prove Hypothesis bj-spim.denote. (Contributed by BJ, 4-Apr-2026.)

Hypotheses

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

Assertion

Ref	Expression
bj-spim	⊢ (𝜑 → (∀𝑥𝜒 → 𝜃))

Proof of Theorem bj-spim

Step	Hyp	Ref	Expression
1		bj-spim.nf	. 2 ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
2		bj-spim.denote	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
3		bj-spim.nf0	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4		bj-spim.maj	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
5	4	ex 417	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
6	3, 5	eximdh 1891	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜒 → 𝜃)))
7	2, 6	mpd 16	. 2 ⊢ (𝜑 → ∃𝑥(𝜒 → 𝜃))
8		bj-spimnfe 37171	. 2 ⊢ ((∃𝑥𝜃 → 𝜃) → (∃𝑥(𝜒 → 𝜃) → (∀𝑥𝜒 → 𝜃)))
9	1, 7, 8	sylc 66	1 ⊢ (𝜑 → (∀𝑥𝜒 → 𝜃))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  bj-cbvalimd0  37175  bj-spim0  37216