Theorem fesapo 2134

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 1524	. . 3 ⊢ (𝜓 → 𝜒)
4		fesapo.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
5	4	spi 1524	. . . 4 ⊢ (𝜑 → ¬ 𝜓)
6	5	con2i 617	. . 3 ⊢ (𝜓 → ¬ 𝜑)
7	3, 6	jca 304	. 2 ⊢ (𝜓 → (𝜒 ∧ ¬ 𝜑))
8	1, 7	eximii 1590	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)