Theorem disamis 2314

Description: "Disamis", one of the syllogisms of Aristotelian logic. Some φ is ψ, and all φ is χ, therefore some χ is ψ. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
disamis	⊢ ∃x(χ ∧ ψ)

Proof of Theorem disamis

Step	Hyp	Ref	Expression
1		disamis.maj	. 2 ⊢ ∃x(φ ∧ ψ)
2		disamis.min	. . . . 5 ⊢ ∀x(φ → χ)
3	2	spi 1753	. . . 4 ⊢ (φ → χ)
4	3	anim1i 551	. . 3 ⊢ ((φ ∧ ψ) → (χ ∧ ψ))
5	4	eximi 1576	. 2 ⊢ (∃x(φ ∧ ψ) → ∃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:  bocardo  2316