Theorem exlimimdd 2220

Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2221. (Revised by Wolf Lammen, 3-Sep-2023.)

Hypotheses

Ref	Expression
exlimdd.1	⊢ Ⅎ𝑥𝜑
exlimdd.2	⊢ Ⅎ𝑥𝜒
exlimdd.3	⊢ (𝜑 → ∃𝑥𝜓)
exlimimdd.4	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
exlimimdd	⊢ (𝜑 → 𝜒)

Proof of Theorem exlimimdd

Step	Hyp	Ref	Expression
1		exlimdd.3	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		exlimdd.1	. . 3 ⊢ Ⅎ𝑥𝜑
3		exlimdd.2	. . 3 ⊢ Ⅎ𝑥𝜒
4		exlimimdd.4	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	2, 3, 4	exlimd 2219	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  exlimdd  2221  tz6.12cOLD  6908  ovmpodf  7568  gsum2d2lem  19959  2ndresdju  32632  padct  32702  stoweidlem27  46023  intsaluni  46325  isomenndlem  46526  tz6.12c-afv2  47238