Theorem exlimimdd 33191

Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)

Hypotheses

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

Assertion

Ref	Expression
exlimimdd	⊢ (𝜑 → 𝜒)

Proof of Theorem exlimimdd

Step	Hyp	Ref	Expression
1		exlimimdd.1	. 2 ⊢ Ⅎ𝑥𝜑
2		exlimimdd.2	. 2 ⊢ Ⅎ𝑥𝜒
3		exlimimdd.3	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
4		exlimimdd.4	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	4	imp 445	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
6	1, 2, 3, 5	exlimdd 2088	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)