Theorem exlimdd 2218

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 416	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	1, 2, 3, 5	exlimimdd 2217	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781  Ⅎwnf 1785

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-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786

This theorem is referenced by:  exlimimddOLD  2220  fvmptd3f  6760  ovmpodf  7285  ex-natded9.26  28204  suprnmpt  41798  stoweidlem43  42685  stoweidlem44  42686  stoweidlem54  42696