Theorem disamis 2198

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 1589	. . 3 ⊢ (𝜑 → 𝜒)
4	3	anim1i 340	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1655	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wal 1400  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  bocardo  2200