P. 444 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44301-44400   *Has distinct variable group(s)

Type	Label	Description
Statement
21.39.5  Calculus
Theorem	lhe4.4ex1a 44301	Example of the Fundamental Theorem of Calculus, part two (ftc2 26001): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 26001 as simply the "Fundamental Theorem of Calculus", then ftc1 25999 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
⊢ ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3)
Theorem	dvsconst 44302	Derivative of a constant function on the real or complex numbers. The function may return a complex 𝐴 even if 𝑆 is ℝ. (Contributed by Steve Rodriguez, 11-Nov-2015.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐴 ∈ ℂ) → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0}))
Theorem	dvsid 44303	Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.)
⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D ( I ↾ 𝑆)) = (𝑆 × {1}))
Theorem	dvsef 44304	Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.)
⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (exp ↾ 𝑆)) = (exp ↾ 𝑆))
Theorem	expgrowthi 44305*	Exponential growth and decay model. See expgrowth 44307 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐾 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝑌 = (𝑡 ∈ 𝑆 ↦ (𝐶 · (exp‘(𝐾 · 𝑡))))    ⇒   ⊢ (𝜑 → (𝑆 D 𝑌) = ((𝑆 × {𝐾}) ∘f · 𝑌))
Theorem	dvconstbi 44306*	The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 25868 and dveq0 25955. Corresponds to integration formula "∫0 d𝑥 = 𝐶 " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑌:𝑆⟶ℂ)    &   ⊢ (𝜑 → dom (𝑆 D 𝑌) = 𝑆)    ⇒   ⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑆 × {𝑐})))
Theorem	expgrowth 44307*	Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 44305 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives. Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model 44305); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ. Here y' is given as (𝑆 D 𝑌), C as 𝑐, and ky as ((𝑆 × {𝐾}) ∘f · 𝑌). (𝑆 × {𝐾}) is the constant function that maps any real or complex input to k and ∘f · is multiplication as a function operation. The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf 44305 pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case 44305. Statements for this and expgrowthi 44305 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐾 ∈ ℂ)    &   ⊢ (𝜑 → 𝑌:𝑆⟶ℂ)    &   ⊢ (𝜑 → dom (𝑆 D 𝑌) = 𝑆)    ⇒   ⊢ (𝜑 → ((𝑆 D 𝑌) = ((𝑆 × {𝐾}) ∘f · 𝑌) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑡 ∈ 𝑆 ↦ (𝑐 · (exp‘(𝐾 · 𝑡))))))
21.39.6  The generalized binomial coefficient operation
Syntax	cbcc 44308	Extend class notation to include the generalized binomial coefficient operation.
class C𝑐
Definition	df-bcc 44309*	Define a generalized binomial coefficient operation, which unlike df-bc 14319 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
Theorem	bccval 44310	Value of the generalized binomial coefficient, 𝐶 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
Theorem	bcccl 44311	Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ)
Theorem	bcc0 44312	The generalized binomial coefficient 𝐶 choose 𝐾 is zero iff 𝐶 is an integer between zero and (𝐾 − 1) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐶C𝑐𝐾) = 0 ↔ 𝐶 ∈ (0...(𝐾 − 1))))
Theorem	bccp1k 44313	Generalized binomial coefficient: 𝐶 choose (𝐾 + 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐶C𝑐(𝐾 + 1)) = ((𝐶C𝑐𝐾) · ((𝐶 − 𝐾) / (𝐾 + 1))))
Theorem	bccm1k 44314	Generalized binomial coefficient: 𝐶 choose (𝐾 − 1), when 𝐶 is not (𝐾 − 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}))    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾)))
Theorem	bccn0 44315	Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶C𝑐0) = 1)
Theorem	bccn1 44316	Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶C𝑐1) = 𝐶)
Theorem	bccbc 44317	The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾))
21.39.7  Binomial series
Theorem	uzmptshftfval 44318*	When 𝐹 is a maps-to function on some set of upper integers 𝑍 that returns a set 𝐵, (𝐹 shift 𝑁) is another maps-to function on the shifted set of upper integers 𝑊. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝑍 ↦ 𝐵)    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶))
Theorem	dvradcnv2 44319*	The radius of convergence of the (formal) derivative 𝐻 of the power series 𝐺 is (at least) as large as the radius of convergence of 𝐺. This version of dvradcnv 26380 uses a shifted version of 𝐻 to match the sum form of (ℂ D 𝐹) in pserdv2 26390 (and shows how to use uzmptshftfval 44318 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝑛 · (𝐴‘𝑛)) · (𝑋↑(𝑛 − 1))))    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝑋) < 𝑅)    ⇒   ⊢ (𝜑 → seq1( + , 𝐻) ∈ dom ⇝ )
Theorem	binomcxplemwb 44320	Lemma for binomcxp 44329. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (((𝐶 − 𝐾) · (𝐶C𝑐𝐾)) + ((𝐶 − (𝐾 − 1)) · (𝐶C𝑐(𝐾 − 1)))) = (𝐶 · (𝐶C𝑐𝐾)))
Theorem	binomcxplemnn0 44321*	Lemma for binomcxp 44329. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15844 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set (0...𝐶), and when the index set is widened beyond 𝐶 the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴↑𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
Theorem	binomcxplemrat 44322*	Lemma for binomcxp 44329. As 𝑘 increases, this ratio's absolute value converges to one. Part of equation "Since continuity of the absolute value..." in the Wikibooks proof (proven for the inverse ratio, which we later show is no problem). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐶 − 𝑘) / (𝑘 + 1)))) ⇝ 1)
Theorem	binomcxplemfrat 44323*	Lemma for binomcxp 44329. binomcxplemrat 44322 implies that when 𝐶 is not a nonnegative integer, the absolute value of the ratio ((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)))) ⇝ 1)
Theorem	binomcxplemradcnv 44324*	Lemma for binomcxp 44329. By binomcxplemfrat 44323 and radcnvrat 44286 the radius of convergence of power series Σ𝑘 ∈ ℕ0((𝐹‘𝑘) · (𝑏↑𝑘)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = 1)
Theorem	binomcxplemdvbinom 44325*	Lemma for binomcxp 44329. By the power and chain rules, calculate the derivative of ((1 + 𝑏)↑𝑐-𝐶), with respect to 𝑏 in the disk of convergence 𝐷. We later multiply the derivative in the later binomcxplemdvsum 44327 by this derivative to show that ((1 + 𝑏)↑𝑐𝐶) (with a nonnegated 𝐶) and the later sum, since both at 𝑏 = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (ℂ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑𝑐-𝐶))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑𝑐(-𝐶 − 1)))))
Theorem	binomcxplemcvg 44326*	Lemma for binomcxp 44329. The sum in binomcxplemnn0 44321 and its derivative (see the next theorem, binomcxplemdvsum 44327) converge, as long as their base 𝐽 is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (seq0( + , (𝑆‘𝐽)) ∈ dom ⇝ ∧ seq1( + , (𝐸‘𝐽)) ∈ dom ⇝ ))
Theorem	binomcxplemdvsum 44327*	Lemma for binomcxp 44329. The derivative of the generalized sum in binomcxplemnn0 44321. Part of remark "This convergence allows to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    &   ⊢ 𝑃 = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘))    ⇒   ⊢ (𝜑 → (ℂ D 𝑃) = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘)))
Theorem	binomcxplemnotnn0 44328*	Lemma for binomcxp 44329. When 𝐶 is not a nonnegative integer, the generalized sum in binomcxplemnn0 44321 —which we will call 𝑃 —is a convergent power series: its base 𝑏 is always of smaller absolute value than the radius of convergence. pserdv2 26390 gives the derivative of 𝑃, which by dvradcnv 26380 also converges in that radius. When 𝐴 is fixed at one, (𝐴 + 𝑏) times that derivative equals (𝐶 · 𝑃) and fraction (𝑃 / ((𝐴 + 𝑏)↑𝑐𝐶)) is always defined with derivative zero, so the fraction is a constant—specifically one, because ((1 + 0)↑𝑐𝐶) = 1. Thus ((1 + 𝑏)↑𝑐𝐶) = (𝑃‘𝑏). Finally, let 𝑏 be (𝐵 / 𝐴), and multiply both the binomial ((1 + (𝐵 / 𝐴))↑𝑐𝐶) and the sum (𝑃‘(𝐵 / 𝐴)) by (𝐴↑𝑐𝐶) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    &   ⊢ 𝑃 = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴↑𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
Theorem	binomcxp 44329*	Generalize the binomial theorem binom 15844 to positive real summand 𝐴, real summand 𝐵, and complex exponent 𝐶. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15844; see also https://en.wikipedia.org/wiki/Binomial_series 15844, https://en.wikipedia.org/wiki/Binomial_theorem 15844 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15844. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴↑𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
21.40  Mathbox for Andrew Salmon
21.40.1  Principia Mathematica * 10
Theorem	pm10.12 44330*	Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥𝜓))
Theorem	pm10.14 44331	Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Theorem	pm10.251 44332	Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	pm10.252 44333	Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
Theorem	pm10.253 44334	Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑)
Theorem	albitr 44335	Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥(𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜓 ↔ 𝜒)) → ∀𝑥(𝜑 ↔ 𝜒))
Theorem	pm10.42 44336	Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∨ 𝜓))
Theorem	pm10.52 44337*	Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓))
Theorem	pm10.53 44338	Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
Theorem	pm10.541 44339*	Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜒 ∨ 𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.542 44340*	Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.55 44341	Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∃𝑥(𝜑 ∧ 𝜓) ∧ ∀𝑥(𝜑 → 𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.56 44342	Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒))
Theorem	pm10.57 44343	Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒)))
21.40.2  Principia Mathematica * 11
Theorem	2alanimi 44344	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦𝜒)
Theorem	2al2imi 44345	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	pm11.11 44346	Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ 𝜑    ⇒   ⊢ ∀𝑧∀𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
Theorem	pm11.12 44347*	Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))
Theorem	19.21vv 44348*	Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1939. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑))
Theorem	2alim 44349	Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	2albi 44350	Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	2exim 44351	Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	2exbi 44352	Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	spsbce-2 44353	Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	19.33-2 44354	Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))
Theorem	19.36vv 44355*	Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓))
Theorem	19.31vv 44356*	Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))
Theorem	19.37vv 44357*	Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))
Theorem	19.28vv 44358*	Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∀𝑥∀𝑦𝜑))
Theorem	pm11.52 44359	Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓))
Theorem	aaanv 44360*	Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2332. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))
Theorem	pm11.57 44361*	Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Theorem	pm11.58 44362*	Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Theorem	pm11.59 44363*	Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
Theorem	pm11.6 44364*	Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	pm11.61 44365*	Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
Theorem	pm11.62 44366*	Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒)))
Theorem	pm11.63 44367	Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))
Theorem	pm11.7 44368	Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
Theorem	pm11.71 44369*	Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))))
21.40.3  Predicate Calculus
Theorem	sbeqal1 44370*	If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)
Theorem	sbeqal1i 44371*	Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)    ⇒   ⊢ 𝑦 = 𝑧
Theorem	sbeqal2i 44372*	If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)    ⇒   ⊢ 𝑧 = 𝑦
Theorem	axc5c4c711 44373	Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1795 as the inference rule. This proof extends the idea of axc5c711 38882 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))
Theorem	axc5c4c711toc5 44374	Rederivation of sp 2183 from axc5c4c711 44373. Note that ax6 2388 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1968 instead of ax6 2388, so that this rederivation requires only ax6v 1968 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axc5c4c711toc4 44375	Rederivation of axc4 2321 from axc5c4c711 44373. Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	axc5c4c711toc7 44376	Rederivation of axc7 2317 from axc5c4c711 44373. Note that neither axc7 2317 nor ax-11 2157 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc5c4c711to11 44377	Rederivation of ax-11 2157 from axc5c4c711 44373. Note that ax-11 2157 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	axc11next 44378*	This theorem shows that, given axextb 2710, we can derive a version of axc11n 2430. However, it is weaker than axc11n 2430 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
21.40.4  Principia Mathematica * 13 and * 14
Theorem	pm13.13a 44379	One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)
Theorem	pm13.13b 44380	Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑)
Theorem	pm13.14 44381	Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)
Theorem	pm13.192 44382*	Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Theorem	pm13.193 44383	Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦))
Theorem	pm13.194 44384	Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))
Theorem	pm13.195 44385*	Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3793. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Theorem	pm13.196a 44386*	Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (¬ 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥))
Theorem	2sbc6g 44387*	Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Theorem	2sbc5g 44388*	Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Theorem	iotain 44389	Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
Theorem	iotaexeu 44390	The iota class exists. This theorem does not require ax-nul 5276 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
Theorem	iotasbc 44391*	Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define ℩ in terms of a function of (℩𝑥𝜑). Their definition differs in that a function of (℩𝑥𝜑) evaluates to "false" when there isn't a single 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))
Theorem	iotasbc2 44392*	Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒)))
Theorem	pm14.12 44393*	Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	pm14.122a 44394*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
Theorem	pm14.122b 44395*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Theorem	pm14.122c 44396*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Theorem	pm14.123a 44397*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
Theorem	pm14.123b 44398*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))
Theorem	pm14.123c 44399*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))
Theorem	pm14.18 44400	Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))