P. 376 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 37501-37600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ispridl 37501*	The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))

Theorem	pridlidl 37502	A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))

Theorem	pridlnr 37503	A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ≠ 𝑋)

Theorem	pridl 37504*	The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐻 = (2^nd ‘𝑅) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))

Theorem	ispridl2 37505*	A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 37537 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))

Theorem	maxidlval 37506*	The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})

Theorem	ismaxidl 37507*	The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))))

Theorem	maxidlidl 37508	A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))

Theorem	maxidlnr 37509	A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋)

Theorem	maxidlmax 37510	A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))

Theorem	maxidln1 37511	One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)

Theorem	maxidln0 37512	A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

21.22.21 Prime rings and integral domains

Syntax	cprrng 37513	Extend class notation with the class of prime rings.
class PrRing

Syntax	cdmn 37514	Extend class notation with the class of domains.
class Dmn

Definition	df-prrngo 37515	Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}

Definition	df-dmn 37516	Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ Dmn = (PrRing ∩ Com2)

Theorem	isprrngo 37517	The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))

Theorem	prrngorngo 37518	A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Theorem	smprngopr 37519	A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Theorem	divrngpr 37520	A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Theorem	isdmn 37521	The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Theorem	isdmn2 37522	The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))

Theorem	dmncrng 37523	A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Theorem	dmnrngo 37524	A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Theorem	flddmn 37525	A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn)

21.22.22 Ideal generators

Syntax	cigen 37526	Extend class notation with the ideal generation function.
class IdlGen

Definition	df-igen 37527*	Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1^st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})

Theorem	igenval 37528*	The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})

Theorem	igenss 37529	A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Theorem	igenidl 37530	The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))

Theorem	igenmin 37531	The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)

Theorem	igenidl2 37532	The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Theorem	igenval2 37533*	The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))

Theorem	prnc 37534*	A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})

Theorem	isfldidl 37535	Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝐾) & ⊢ 𝐻 = (2^nd ‘𝐾) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Theorem	isfldidl2 37536	Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝐾) & ⊢ 𝐻 = (2^nd ‘𝐾) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Theorem	ispridlc 37537*	The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))

Theorem	pridlc 37538	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

Theorem	pridlc2 37539	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)

Theorem	pridlc3 37540	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))

Theorem	isdmn3 37541*	The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))

Theorem	dmnnzd 37542	A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))

Theorem	dmncan1 37543	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))

Theorem	dmncan2 37544	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))

21.23 Mathbox for Giovanni Mascellani

21.23.1 Tools for automatic proof building The results in this section are mostly meant for being used by automatic proof building programs. As a result, they might appear less useful or meaningful than others to human beings.

Theorem	efald2 37545	A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ 𝜑 → ⊥) ⇒ ⊢ 𝜑

Theorem	notbinot1 37546	Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ (¬ 𝜑 ↔ 𝜓) ↔ (𝜑 ↔ 𝜓))

Theorem	bicontr 37547	Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥)

Theorem	impor 37548	An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒))

Theorem	orfa 37549	The falsum ⊥ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((𝜑 ∨ ⊥) ↔ 𝜑)

Theorem	notbinot2 37550	Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ (𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓))

Theorem	biimpor 37551	A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ↔ 𝜓) ∨ 𝜒))

Theorem	orfa1 37552	Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ ⊥) → 𝜓)

Theorem	orfa2 37553	Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ⊥) ⇒ ⊢ ((𝜑 ∨ 𝜓) → 𝜓)

Theorem	bifald 37554	Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → (𝜓 ↔ ⊥))

Theorem	orsild 37555	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → ¬ 𝜓)

Theorem	orsird 37556	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → ¬ 𝜒)

Theorem	cnf1dd 37557	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒)) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃))

Theorem	cnf2dd 37558	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ¬ 𝜃)) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒))

Theorem	cnfn1dd 37559	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → (¬ 𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃))

Theorem	cnfn2dd 37560	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒))

Theorem	or32dd 37561	A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃) ∨ 𝜏))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) ∨ 𝜃)))

Theorem	notornotel1 37562	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓)

Theorem	notornotel2 37563	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → ¬ (𝜓 ∨ ¬ 𝜒)) ⇒ ⊢ (𝜑 → 𝜒)

Theorem	contrd 37564	A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒)) & ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓)

Theorem	an12i 37565	An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
⊢ (𝜑 ∧ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 ∧ (𝜑 ∧ 𝜒))

Theorem	exmid2 37566	An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ ((𝜓 ∧ 𝜑) → 𝜒) & ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜂) → 𝜒)

Theorem	selconj 37567	An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒)))

Theorem	truconj 37568	Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ (𝜑 ↔ (⊤ ∧ 𝜑))

Theorem	orel 37569	An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ ((𝜓 ∧ 𝜂) → 𝜃) & ⊢ ((𝜒 ∧ 𝜌) → 𝜃) & ⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜃)

Theorem	negel 37570	An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜓 → 𝜒) & ⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ⊥)

Theorem	botel 37571	An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜑 → ⊥) ⇒ ⊢ (𝜑 → 𝜓)

Theorem	tradd 37572	Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ (⊤ ∧ 𝜓))

Theorem	gm-sbtru 37573	Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)

Theorem	sbfal 37574	Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥)

Theorem	sbcani 37575	Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) & ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) ⇒ ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜂))

Theorem	sbcori 37576	Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) & ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) ⇒ ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂))

Theorem	sbcimi 37577	Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) & ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) ⇒ ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Theorem	sbcni 37578	Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Theorem	sbali 37579	Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Theorem	sbexi 37580	Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Theorem	sbcalf 37581*	Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)

Theorem	sbcexf 37582*	Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)

Theorem	sbcalfi 37583*	Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ Ⅎ𝑦𝐴 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)

Theorem	sbcexfi 37584*	Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ Ⅎ𝑦𝐴 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓)

Theorem	spsbcdi 37585	A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ 𝐴 ∈ V & ⊢ (𝜑 → ∀𝑥𝜒) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 → 𝜓)

Theorem	alrimii 37586*	A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝜓) & ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓) & ⊢ Ⅎ𝑦𝜒 ⇒ ⊢ (𝜑 → ∀𝑥𝜒)

Theorem	spesbcdi 37587	A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ (𝜑 → 𝜓) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜒)

Theorem	exlimddvf 37588	A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑥𝜓 & ⊢ ((𝜃 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem	exlimddvfi 37589	A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑦𝜃 & ⊢ Ⅎ𝑦𝜓 & ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂) & ⊢ ((𝜂 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑦𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem	sbceq1ddi 37590	A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝜃) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜃) & ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜂)

Theorem	sbccom2lem 37591*	Lemma for sbccom2 37592. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theorem	sbccom2 37592*	Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theorem	sbccom2f 37593*	Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theorem	sbccom2fi 37594*	Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓)

Theorem	csbcom2fi 37595*	Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 & ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸 ⇒ ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸

21.23.2 Tseitin axioms A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form.

Theorem	fald 37596	Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ¬ ⊥)

Theorem	tsim1 37597	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)))

Theorem	tsim2 37598	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 → 𝜓)))

Theorem	tsim3 37599	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 → 𝜓)))

Theorem	tsbi1 37600	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))

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