Theorem fresison 2117

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

Hypotheses

Ref	Expression
fresison.maj	⊢ ∀𝑥(𝜑 → ¬ 𝜓)
fresison.min	⊢ ∃𝑥(𝜓 ∧ 𝜒)

Assertion

Ref	Expression
fresison	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fresison

Step	Hyp	Ref	Expression
1		fresison.min	. 2 ⊢ ∃𝑥(𝜓 ∧ 𝜒)
2		simpr 109	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
3		fresison.maj	. . . . . 6 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
4	3	spi 1516	. . . . 5 ⊢ (𝜑 → ¬ 𝜓)
5	4	con2i 616	. . . 4 ⊢ (𝜓 → ¬ 𝜑)
6	5	adantr 274	. . 3 ⊢ ((𝜓 ∧ 𝜒) → ¬ 𝜑)
7	2, 6	jca 304	. 2 ⊢ ((𝜓 ∧ 𝜒) → (𝜒 ∧ ¬ 𝜑))
8	1, 7	eximii 1581	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329  ∃wex 1468

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)