Theorem bamalip 2324

Description: "Bamalip", one of the syllogisms of Aristotelian logic. All φ is ψ, all ψ is χ, and φ exist, therefore some χ is φ. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2305. (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

Ref	Expression
bamalip.maj	⊢ ∀x(φ → ψ)
bamalip.min	⊢ ∀x(ψ → χ)
bamalip.e	⊢ ∃xφ

Assertion

Ref	Expression
bamalip	⊢ ∃x(χ ∧ φ)

Proof of Theorem bamalip

Step	Hyp	Ref	Expression
1		bamalip.e	. 2 ⊢ ∃xφ
2		bamalip.maj	. . . . . 6 ⊢ ∀x(φ → ψ)
3	2	spi 1753	. . . . 5 ⊢ (φ → ψ)
4		bamalip.min	. . . . . 6 ⊢ ∀x(ψ → χ)
5	4	spi 1753	. . . . 5 ⊢ (ψ → χ)
6	3, 5	syl 15	. . . 4 ⊢ (φ → χ)
7	6	ancri 535	. . 3 ⊢ (φ → (χ ∧ φ))
8	7	eximi 1576	. 2 ⊢ (∃xφ → ∃x(χ ∧ φ))
9	1, 8	ax-mp 5	1 ⊢ ∃x(χ ∧ φ)

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: (None)