Theorem darii 2119

Description: "Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)

Hypotheses

Ref	Expression
darii.maj	⊢ ∀𝑥(𝜑 → 𝜓)
darii.min	⊢ ∃𝑥(𝜒 ∧ 𝜑)

Assertion

Ref	Expression
darii	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem darii

Step	Hyp	Ref	Expression
1		darii.min	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑)
2		darii.maj	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜓)
3	2	spi 1529	. . 3 ⊢ (𝜑 → 𝜓)
4	3	anim2i 340	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1595	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ferio  2120