Theorem exlimddvf 38495

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 414	. . 3 ⊢ (𝜓 → (𝜃 → 𝜒))
6	2, 3, 5	exlimd 2230	. 2 ⊢ (𝜓 → (∃𝑥𝜃 → 𝜒))
7	1, 6	mpan9 511	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791

This theorem is referenced by:  exlimddvfi  38496