Theorem exlimdd 2255

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)

Hypotheses

Ref	Expression
exlimdd.1	⊢ Ⅎ𝑥𝜑
exlimdd.2	⊢ Ⅎ𝑥𝜒
exlimdd.3	⊢ (𝜑 → ∃𝑥𝜓)
exlimdd.4	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
exlimdd	⊢ (𝜑 → 𝜒)

Proof of Theorem exlimdd

Step	Hyp	Ref	Expression
1		exlimdd.1	. 2 ⊢ Ⅎ𝑥𝜑
2		exlimdd.2	. 2 ⊢ Ⅎ𝑥𝜒
3		exlimdd.3	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
4		exlimdd.4	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
5	4	ex 417	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	1, 2, 3, 5	exlimimdd 2254	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∃wex 1808  Ⅎwnf 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813

This theorem is used by:  fvmptd3f  7005  ovmpodf  7568  ex-natded9.26  30781  stoweidlem43  46785  stoweidlem44  46786  stoweidlem54  46796