Theorem exlimimd 38030

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

Hypotheses

Ref	Expression
exlimimd.1	⊢ (𝜑 → ∃𝑥𝜓)
exlimimd.2	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
exlimimd	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem exlimimd

Step	Hyp	Ref	Expression
1		exlimimd.1	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		exlimimd.2	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imp 412	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
4	1, 3	exlimddv 1968	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)