Theorem disamis 2564

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	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜒)
3	2	spi 2042	. . 3 ⊢ (𝜑 → 𝜒)
4	3	anim1i 590	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1754	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  bocardo  2566