Theorem bamalip 2021

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 2002. (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	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
3	2	spi 1429	. . . 4 ⊢ (𝜑 → 𝜓)
4		bamalip.min	. . . . 5 ⊢ ∀𝑥(𝜓 → 𝜒)
5	4	spi 1429	. . . 4 ⊢ (𝜓 → 𝜒)
6	3, 5	syl 14	. . 3 ⊢ (𝜑 → 𝜒)
7	6	ancri 307	. 2 ⊢ (𝜑 → (𝜒 ∧ 𝜑))
8	1, 7	eximii 1493	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)