Theorem fesapo 2165

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 1550	. . 3 ⊢ (𝜓 → 𝜒)
4		fesapo.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
5	4	spi 1550	. . . 4 ⊢ (𝜑 → ¬ 𝜓)
6	5	con2i 628	. . 3 ⊢ (𝜓 → ¬ 𝜑)
7	3, 6	jca 306	. 2 ⊢ (𝜓 → (𝜒 ∧ ¬ 𝜑))
8	1, 7	eximii 1616	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)