Theorem exlimdd 2254

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 402	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 3, 5	exlimd 2253	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
7	1, 6	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 385  ∃wex 1875  Ⅎwnf 1879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880

This theorem is referenced by:  exlimimdd  2255  fvmptd3f  6520  ovmpt2df  7026  ex-natded9.26  27804  suprnmpt  40110  stoweidlem43  41003  stoweidlem44  41004  stoweidlem54  41014