Theorem exlimdd 2205

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 412	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	1, 2, 3, 5	exlimimdd 2204	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778

This theorem is referenced by:  fvmptd3f  7006  ovmpodf  7559  ex-natded9.26  30177  stoweidlem43  45312  stoweidlem44  45313  stoweidlem54  45323