Theorem exlimddvf 38811

Description: A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
exlimddvf.1	⊢ (𝜑 → ∃𝑥𝜃)
exlimddvf.2	⊢ Ⅎ𝑥𝜓
exlimddvf.3	⊢ ((𝜃 ∧ 𝜓) → 𝜒)
exlimddvf.4	⊢ Ⅎ𝑥𝜒

Assertion

Ref	Expression
exlimddvf	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem exlimddvf

Step	Hyp	Ref	Expression
1		exlimddvf.1	. 2 ⊢ (𝜑 → ∃𝑥𝜃)
2		exlimddvf.2	. . 3 ⊢ Ⅎ𝑥𝜓
3		exlimddvf.4	. . 3 ⊢ Ⅎ𝑥𝜒
4		exlimddvf.3	. . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒)
5	4	expcom 419	. . 3 ⊢ (𝜓 → (𝜃 → 𝜒))
6	2, 3, 5	exlimd 2257	. 2 ⊢ (𝜓 → (∃𝑥𝜃 → 𝜒))
7	1, 6	mpan9 516	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812  Ⅎwnf 1816

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  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  exlimddvfi  38812