Theorem datisi 2313

Description: "Datisi", one of the syllogisms of Aristotelian logic. All φ is ψ, and some φ is χ, therefore some χ is ψ. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
datisi	⊢ ∃x(χ ∧ ψ)

Proof of Theorem datisi

Step	Hyp	Ref	Expression
1		datisi.min	. 2 ⊢ ∃x(φ ∧ χ)
2		simpr 447	. . . 4 ⊢ ((φ ∧ χ) → χ)
3		datisi.maj	. . . . . 6 ⊢ ∀x(φ → ψ)
4	3	spi 1753	. . . . 5 ⊢ (φ → ψ)
5	4	adantr 451	. . . 4 ⊢ ((φ ∧ χ) → ψ)
6	2, 5	jca 518	. . 3 ⊢ ((φ ∧ χ) → (χ ∧ ψ))
7	6	eximi 1576	. 2 ⊢ (∃x(φ ∧ χ) → ∃x(χ ∧ ψ))
8	1, 7	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:  ferison  2315