Theorem fesapo 2585

Description: "Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
fesapo.maj	⊢ ∀𝑥(𝜑 → ¬ 𝜓)
fesapo.min	⊢ ∀𝑥(𝜓 → 𝜒)
fesapo.e	⊢ ∃𝑥𝜓

Assertion

Ref	Expression
fesapo	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fesapo

Step	Hyp	Ref	Expression
1		fesapo.e	. 2 ⊢ ∃𝑥𝜓
2		fesapo.min	. . . 4 ⊢ ∀𝑥(𝜓 → 𝜒)
3	2	spi 2054	. . 3 ⊢ (𝜓 → 𝜒)
4		fesapo.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
5	4	spi 2054	. . . 4 ⊢ (𝜑 → ¬ 𝜓)
6	5	con2i 134	. . 3 ⊢ (𝜓 → ¬ 𝜑)
7	3, 6	jca 554	. 2 ⊢ (𝜓 → (𝜒 ∧ ¬ 𝜑))
8	1, 7	eximii 1764	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)