Theorem bamalip 2693

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. Very similar to barbari 2670. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

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.min	. . 3 ⊢ ∀𝑥(𝜓 → 𝜒)
2		bamalip.maj	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
3		bamalip.e	. . 3 ⊢ ∃𝑥𝜑
4	1, 2, 3	barbari 2670	. 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒)
5		exancom 1865	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜑))
6	4, 5	mpbi 229	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by: (None)