Theorem exlimdd 1883

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 115	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 3, 5	exlimd 1608	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
7	1, 6	mpd 13	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1471  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie2 1505  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  fvmptdf  5619  ovmpodf  6023  exmidfodomrlemr  7219  exmidfodomrlemrALT  7220  ltexprlemm  7617  dfgrp3mlem  13008