Theorem exlimddv 1638

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypotheses

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

Assertion

Ref	Expression
exlimddv	⊢ (φ → χ)

Distinct variable groups:   χ,x   φ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem exlimddv

Step	Hyp	Ref	Expression
1		exlimddv.1	. 2 ⊢ (φ → ∃xψ)
2		exlimddv.2	. . . 4 ⊢ ((φ ∧ ψ) → χ)
3	2	ex 423	. . 3 ⊢ (φ → (ψ → χ))
4	3	exlimdv 1636	. 2 ⊢ (φ → (∃xψ → χ))
5	1, 4	mpd 14	1 ⊢ (φ → χ)

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

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

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

This theorem is referenced by: (None)