P. 27 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	euor2 2601	Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))
Theorem	sbmo 2602*	Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Theorem	eu4 2603*	Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
Theorem	euimmo 2604	Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
Theorem	euim 2605	Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof shortened by Wolf Lammen, 1-Oct-2023.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
Theorem	moanimlem 2606	Factor out the common proof skeleton of moanimv 2607 and moanim 2608. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) Factor out common proof lines. (Revised by Wolf Lammen, 8-Feb-2023.)
⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))    &   ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)    ⇒   ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moanimv 2607*	Introduction of a conjunct into an at-most-one quantifier. Version of moanim 2608 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce axiom usage . (Revised by Wolf Lammen, 8-Feb-2023.)
⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moanim 2608	Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2607. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	euan 2609	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	moanmo 2610	Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
Theorem	moaneu 2611	Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Theorem	euanv 2612*	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	mopick 2613	"At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	moexexlem 2614	Factor out the proof skeleton of moexex 2626 and moexexvw 2616. (Contributed by Wolf Lammen, 2-Oct-2023.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦∃*𝑥𝜑    &   ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)    ⇒   ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	2moexv 2615*	Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Theorem	moexexvw 2616*	"At most one" double quantification. Version of moexexv 2627 with an additional disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 26-Jan-1997.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2626. (Revised by Wolf Lammen, 2-Oct-2023.)
⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	2moswapv 2617*	A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap 2632 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2616. (Revised by Wolf Lammen, 2-Oct-2023.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
Theorem	2euswapv 2618*	A condition allowing to swap an existential quantifier and a unique existential quantifier. Version of 2euswap 2633 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
Theorem	2euexv 2619*	Double quantification with existential uniqueness. Version of 2euex 2629 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2365. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Theorem	2exeuv 2620*	Double existential uniqueness implies double unique existential quantification. Version of 2exeu 2634 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2365. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Theorem	eupick 2621	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupicka 2622	Version of eupick 2621 with closed formulas. (Contributed by NM, 6-Sep-2008.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
Theorem	eupickb 2623	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))
Theorem	eupickbi 2624	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	mopick2 2625	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1881. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	moexex 2626	"At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the version moexexvw 2616 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2616. (Revised by Wolf Lammen, 2-Oct-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	moexexv 2627*	"At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker moexexvw 2616 when possible. (Contributed by NM, 26-Jan-1997.) (New usage is discouraged.)
⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	2moex 2628	Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker 2moexv 2615 when possible. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Theorem	2euex 2629	Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker 2euexv 2619 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Theorem	2eumo 2630	Nested unique existential quantifier and at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
Theorem	2eu2ex 2631	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	2moswap 2632	A condition allowing to swap an existential quantifier and at at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker 2moswapv 2617 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
Theorem	2euswap 2633	A condition allowing to swap an existential quantifier and a unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker 2euswapv 2618 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
Theorem	2exeu 2634	Double existential uniqueness implies double unique existential quantification. The converse does not hold. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker 2exeuv 2620 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Theorem	2mo2 2635*	Two ways of expressing "there exists at most one ordered pair ⟨𝑥, 𝑦⟩ such that 𝜑(𝑥, 𝑦) holds. Note that this is not equivalent to ∃*𝑥∃*𝑦𝜑. See also 2mo 2636. This is the analogue of 2eu4 2643 for existential uniqueness. (Contributed by Wolf Lammen, 26-Oct-2019.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 3-Jan-2023.)
⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2mo 2636*	Two ways of expressing "there exists at most one ordered pair ⟨𝑥, 𝑦⟩ such that 𝜑(𝑥, 𝑦) holds. See also 2mo2 2635. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2mos 2637*	Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) (Proof shortened by Wolf Lammen, 21-May-2025.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2mosOLD 2638*	Obsolete version of 2mos 2637 as of 21-May-2025. (Contributed by NM, 10-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2eu1 2639	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker 2eu1v 2640 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
Theorem	2eu1v 2640*	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Version of 2eu1 2639 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2365. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
Theorem	2eu2 2641	Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)
⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑))
Theorem	2eu3 2642	Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
Theorem	2eu4 2643*	This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2eu1 2639 for a condition under which the naive definition holds and 2exeu 2634 for a one-way implication. See 2eu5 2644 and 2eu8 2647 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2eu5 2644*	An alternate definition of double existential uniqueness (see 2eu4 2643). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦". (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining ∀𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published. (∃* means "there exists at most one".) (Contributed by NM, 26-Oct-2003.) Avoid ax-13 2365. (Revised by Wolf Lammen, 2-Oct-2023.)
⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2eu6 2645*	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	2eu7 2646	Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 19-Feb-2005.) (New usage is discouraged.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Theorem	2eu8 2647	Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2646. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 20-Feb-2005.) (New usage is discouraged.)
⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))
Theorem	euae 2648*	Two ways to express "exactly one thing exists". To paraphrase the statement and explain the label: there Exists a Unique thing if and only if for All 𝑥, 𝑥 Equals some given (and disjoint) 𝑦. Both sides are false in set theory, see Theorems neutru 35944 and dtru 5433. (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant ⊤. (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies. (Revised by Wolf Lammen, 2-Mar-2023.)
⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	exists1 2649*	Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see Theorem dtru 5433. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.)
⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	exists2 2650	A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 4-Mar-2023.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
1.7  Other axiomatizations related to classical predicate calculus
1.7.1  Aristotelian logic: Assertic syllogisms Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1762) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe. This section models this system (including later refinements). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation. We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable 𝑥. Our translation is essentially identical to the one used in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important". There are two differences: we make the existence criteria explicit, and we use 𝜑, 𝜓, and 𝜒 in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the approach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/ 1762. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no 𝜑 is 𝜓 " are consistently translated as ∀𝑥(𝜑 → ¬ 𝜓). These can also be expressed as ¬ ∃𝑥(𝜑 ∧ 𝜓), per alinexa 1837. We translate "all 𝜑 is 𝜓 " to ∀𝑥(𝜑 → 𝜓), "some 𝜑 is 𝜓 " to ∃𝑥(𝜑 ∧ 𝜓), and "some 𝜑 is not 𝜓 " to ∃𝑥(𝜑 ∧ ¬ 𝜓). It is traditional to use the singular form "is", not the plural form "are", in the generic expressions. By convention the major premise is listed first. In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by more specific constructs such as 𝑥 = 𝐴, 𝑥 ∈ 𝐴, or 𝑥 ⊆ 𝐴. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like 𝑥 ∈ 𝐴 instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion (⊆) and membership (∈); this distinction was not directly made in Aristotle's work. There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such nonexistent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/ 1837. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2657, celaront 2659, cesaro 2666, camestros 2667, felapton 2674, darapti 2672, calemos 2678, fesapo 2679, and bamalip 2680. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically, Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelian logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus. The following twenty-four syllogisms (from barbara 2651 to bamalip 2680) are all proven from { ax-mp 5, ax-1 6, ax-2 7, ax-3 8, ax-gen 1789, ax-4 1803 }, which corresponds in the usual translation to modal logic (a universal (resp. existential) quantifier maps to necessity (resp. possibility)) to the weakest normal modal logic (K). Some proofs could be shortened by using additionally spi 2172 (inference form of sp 2171, which corresponds to the axiom (T) of modal logic), as demonstrated by dariiALT 2654, barbariALT 2658, festinoALT 2663, barocoALT 2665, daraptiALT 2673.
Theorem	barbara 2651	"Barbara", one of the fundamental syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is 𝜓. In Aristotelian notation, AAA-1: MaP and SaM therefore SaP. For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (Socrates is mortal). Russell and Whitehead note that "the syllogism in Barbara is derived from [syl 17]" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1888. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1888, http://plato.stanford.edu/entries/aristotle-logic/ 1888, and https://en.wikipedia.org/wiki/Syllogism 1888. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → 𝜓)
Theorem	celarent 2652	"Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. Instance of barbara 2651. In Aristotelian notation, EAE-1: MeP and SaM therefore SeP. For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism 2651. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
Theorem	darii 2653	"Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. In Aristotelian notation, AII-1: MaP and SiM therefore SiP. For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. See dariiALT 2654 for a shorter proof requiring more axioms. (Contributed by David A. Wheeler, 24-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	dariiALT 2654	Alternate proof of darii 2653, shorter but using more axioms. This shows how the use of spi 2172 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2172 is the inference associated with sp 2171, which corresponds to the axiom (T) of modal logic. (Contributed by David A. Wheeler, 27-Aug-2016.) Added precisions on axiom usage. (Revised by BJ, 27-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferio 2655	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. Instance of darii 2653. In Aristotelian notation, EIO-1: MeP and SiM therefore SoP. For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism 2653. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	barbarilem 2656	Lemma for barbari 2657 and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022.)
⊢ ∃𝑥𝜑    &   ⊢ ∀𝑥(𝜑 → 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 ∧ 𝜓)
Theorem	barbari 2657	"Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-1: MaP and SaM therefore SiP. For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	barbariALT 2658	Alternate proof of barbari 2657, shorter but using more axioms. See comment of dariiALT 2654. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	celaront 2659	"Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. Instance of barbari 2657. In Aristotelian notation, EAO-1: MeP and SaM therefore SoP. For example, given "No reptiles have fur", "All snakes are reptiles", and "Snakes exist", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism 2657. (Contributed by David A. Wheeler, 27-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	cesare 2660	"Cesare", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. In Aristotelian notation, EAE-2: PeM and SaM therefore SeP. Related to celarent 2652. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	camestres 2661	"Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-2: PaM and SeM therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	festino 2662	"Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	festinoALT 2663	Alternate proof of festino 2662, shorter but using more axioms. See comment of dariiALT 2654. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	baroco 2664	"Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, AOO-2: PaM and SoM therefore SoP. For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	barocoALT 2665	Alternate proof of festino 2662, shorter but using more axioms. See comment of dariiALT 2654. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	cesaro 2666	"Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	camestros 2667	"Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, AEO-2: PaM and SeM therefore SoP. For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	datisi 2668	"Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	disamis 2669	"Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferison 2670	"Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of datisi 2668. In Aristotelian notation, EIO-3: MeP and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	bocardo 2671	"Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of disamis 2669. In Aristotelian notation, OAO-3: MoP and MaS therefore SoP. For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	darapti 2672	"Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-3: MaP and MaS therefore SiP. For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	daraptiALT 2673	Alternate proof of darapti 2672, shorter but using more axioms. See comment of dariiALT 2654. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	felapton 2674	"Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. Instance of darapti 2672. In Aristotelian notation, EAO-3: MeP and MaS therefore SoP. For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	calemes 2675	"Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	dimatis 2676	"Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. In Aristotelian notation, IAI-4: PiM and MaS therefore SiP. For example, "Some pets are rabbits", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2653 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
Theorem	fresison 2677	"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.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜓 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	calemos 2678	"Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, AEO-4: PaM and MeS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	fesapo 2679	"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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	bamalip 2680	"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. Very similar to barbari 2657. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
1.7.2  Intuitionistic logic Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 892 or peirce 201. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 846 and df-ex 1774 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm. The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1803, ax-11 2146, ax-gen 1789, ax-7 2003, ax-12 2166, ax-8 2100, ax-9 2108, and ax-5 1905. In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.
Theorem	axia1 2681	Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜑)
Theorem	axia2 2682	Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜓)
Theorem	axia3 2683	'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
Theorem	axin1 2684	'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Theorem	axin2 2685	'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	axio 2686	Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
Theorem	axi4 2687	Specialization (intuitionistic logic axiom ax-4). This is just sp 2171 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axi5r 2688	Converse of axc4 2309 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
Theorem	axial 2689	The setvar 𝑥 is not free in ∀𝑥𝜑 (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	axie1 2690	The setvar 𝑥 is not free in ∃𝑥𝜑 (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	axie2 2691	A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	axi9 2692	Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1963 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	axi10 2693	Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2419 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axi12 2694	Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2375 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by Jim Kingdon, 31-Dec-2017.) Avoid ax-11 2146. (Revised by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Theorem	axbnd 2695	Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2694 are fairly straightforward consequences of axc9 2375. But in intuitionistic logic, it is not easy to add the extra ∀𝑥 to axi12 2694 and so we treat the two as separate axioms. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof shortened by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets". A set can be an element of another set, and this relationship is indicated by the ∈ symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's Paradox" (ru 3769), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here. Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties". (The axiom of extensionality can also be restated this way as shown by axexte 2697.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems.
2.1  ZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2696*	Axiom of extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. Its converse is a theorem of predicate logic, elequ2g 2114. Set theory can also be formulated with a single primitive predicate ∈ on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)), and equality 𝑥 = 𝑦 is defined as ∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's predicate calculus axioms, we would rewrite all axioms involving equality with equality expanded according to the above definition. Some of those axioms may be provable from ax-ext and would become redundant, but this hasn't been studied carefully. General remarks: Our set theory axioms are presented using defined connectives (↔, ∃, etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives →, ¬, ∀, =, and ∈. It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable 𝑥 in ax-ext 2696 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both 𝑥 and 𝑧. This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 5281, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2696 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axexte 2697*	The axiom of extensionality (ax-ext 2696) restated so that it postulates the existence of a set 𝑧 given two arbitrary sets 𝑥 and 𝑦. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axextg 2698*	A generalization of the axiom of extensionality in which 𝑥 and 𝑦 need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 2129, ax-12 2166, ax-13 2365. (Revised by BJ, 12-Jul-2019.) (Revised by Wolf Lammen, 9-Dec-2019.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axextb 2699*	A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext 2696 and df-cleq 2717. (Contributed by NM, 14-Nov-2008.)
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	axextmo 2700*	There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)