Theorem exlimdd 1889

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

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.3	. 2 ⊢ (φ → ∃xψ)
2		exlimdd.1	. . 3 ⊢ Ⅎxφ
3		exlimdd.2	. . 3 ⊢ Ⅎxχ
4		exlimdd.4	. . . 4 ⊢ ((φ ∧ ψ) → χ)
5	4	ex 423	. . 3 ⊢ (φ → (ψ → χ))
6	2, 3, 5	exlimd 1806	. 2 ⊢ (φ → (∃xψ → χ))
7	1, 6	mpd 14	1 ⊢ (φ → χ)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)