Theorem darii 2303

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	. . . . 5 ⊢ ∀x(φ → ψ)
3	2	spi 1753	. . . 4 ⊢ (φ → ψ)
4	3	anim2i 552	. . 3 ⊢ ((χ ∧ φ) → (χ ∧ ψ))
5	4	eximi 1576	. 2 ⊢ (∃x(χ ∧ φ) → ∃x(χ ∧ ψ))
6	1, 5	ax-mp 5	1 ⊢ ∃x(χ ∧ ψ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃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  ax-9 1654  ax-8 1675  ax-11 1746

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

This theorem is referenced by:  ferio  2304