P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nadddilem2 36801*	Lemma for nadddi 36804. Prove the reverse implication. (Contributed by Scott Fenton, 31-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))    ⇒   ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ⊆ (𝐴 ·no (𝐵 +no 𝐶)))
Theorem	nadddilem3 36802*	Lemma for nadddi 36804. Prove a subcase of the forward implication. (Contributed by Scott Fenton, 3-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 +no 𝐶))    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ⊆ (𝑍 +no 𝐶))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))    ⇒   ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)))
Theorem	nadddilem4 36803*	Lemma for nadddi 36804. Prove the forward implication. (Contributed by Scott Fenton, 3-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))    ⇒   ⊢ (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
Theorem	nadddi 36804	Natural multiplication distributes over natural addition. (Contributed by Scott Fenton, 27-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
Theorem	nadddid 36805	Natural multiplication distributes over natural addition. Deduction form. (Contributed by Scott Fenton, 3-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
Theorem	nadddird 36806	Natural multiplication distributes over natural addition. Deduction form. (Contributed by Scott Fenton, 3-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    ⇒   ⊢ (𝜑 → ((𝐴 +no 𝐵) ·no 𝐶) = ((𝐴 ·no 𝐶) +no (𝐵 ·no 𝐶)))
21.12  Mathbox for Gino Giotto
21.12.1  Equality theorems
21.12.1.1  Inference versions
Theorem	rmoeqi 36807	Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)
Theorem	rmoeqbii 36808	Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)
Theorem	reueqi 36809	Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)
Theorem	reueqbii 36810	Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)
Theorem	sbceqbii 36811	Formula-building inference for class substitution. General version of sbcbii 3795. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓)
Theorem	disjeq1i 36812	Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)
Theorem	disjeq12i 36813	Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)
Theorem	rabeqbii 36814	Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}
Theorem	iuneq12i 36815	Equality theorem for indexed union. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷
Theorem	iineq1i 36816	Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶
Theorem	iineq12i 36817	Equality theorem for indexed intersection. Inference version. General version of iineq1i 36816. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷
Theorem	riotaeqbii 36818	Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓)
Theorem	riotaeqi 36819	Equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑)
Theorem	ixpeq1i 36820	Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶
Theorem	ixpeq12i 36821	Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷
Theorem	sumeq2si 36822	Equality inference for sum. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	sumeq12si 36823	Equality inference for sum. General version of sumeq2si 36822. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ Σ𝑥 ∈ 𝐴 𝐶 = Σ𝑥 ∈ 𝐵 𝐷
Theorem	prodeq2si 36824	Equality inference for product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶
Theorem	prodeq12si 36825	Equality inference for product. General version of prodeq2si 36824. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
Theorem	itgeq12i 36826	Equality inference for an integral. General version of itgeq1i 36827 and itgeq2i 36828. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥
Theorem	itgeq1i 36827	Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥
Theorem	itgeq2i 36828	Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥
Theorem	ditgeq123i 36829	Equality inference for the directed integral. General version of ditgeq12i 36830 and ditgeq3i 36831. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    &   ⊢ 𝐸 = 𝐹    ⇒   ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑥
Theorem	ditgeq12i 36830	Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥
Theorem	ditgeq3i 36831	Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐶 = 𝐷    ⇒   ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥
21.12.1.2  Deduction versions
Theorem	rmoeqdv 36832*	Formula-building rule for restricted at-most-one quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
Theorem	rmoeqbidv 36833*	Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3380. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒))
Theorem	sbequbidv 36834*	Deduction substituting both sides of a biconditional. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝑢 = 𝑣)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑢 / 𝑥]𝜓 ↔ [𝑣 / 𝑥]𝜒))
Theorem	disjeq12dv 36835*	Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))
Theorem	ixpeq12dv 36836*	Equality theorem for infinite Cartesian product. Deduction version. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷)
Theorem	sumeq12sdv 36837*	Equality deduction for sum. General version of sumeq2sdv 15790. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)
Theorem	prodeq12sdv 36838*	Equality deduction for product. General version of prodeq2sdv 16011. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷)
Theorem	itgeq12sdv 36839*	Equality theorem for an integral. Deduction form. General version of itgeq1d 46785 and itgeq2sdv 36840. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
Theorem	itgeq2sdv 36840*	Equality theorem for an integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
Theorem	ditgeq123dv 36841*	Equality theorem for the directed integral. Deduction form. General version of ditgeq3sdv 36843. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐸 = 𝐹)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑥)
Theorem	ditgeq12d 36842*	Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
Theorem	ditgeq3sdv 36843*	Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)
21.12.2  Change bound variables
Theorem	in-ax8 36844	A proof of ax-8 2147 that does not rely on ax-8 2147. It employs df-in 3906 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2155. Since the nature of this result is unclear, usage of this theorem is discouraged, and this method should not be applied to eliminate axiom dependencies. (Contributed by GG, 1-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Theorem	ss-ax8 36845	A proof of ax-8 2147 that does not rely on ax-8 2147. It employs df-ss 3916 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2155. Contrary to in-ax8 36844, this proof does not rely on df-cleq 2752, therefore using fewer axioms . This method should not be applied to eliminate axiom dependencies. (Contributed by GG, 30-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
21.12.2.1  Change bound variables and domains
Theorem	cbvralvw2 36846*	Change bound variable and domain in the restricted universal quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrexvw2 36847*	Change bound variable and domain in the restricted existential quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrmovw2 36848*	Change bound variable and domain in the restricted at-most-one quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐵 𝜓)
Theorem	cbvreuvw2 36849*	Change bound variable and domain in the restricted existential uniqueness quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)
Theorem	cbvsbcvw2 36850*	Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3773. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓)
Theorem	cbvcsbvw2 36851*	Change bound variable of a proper substitution into a class using implicit substitution. General version of cbvcsbv 3859. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷
Theorem	cbviunvw2 36852*	Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷
Theorem	cbviinvw2 36853*	Change bound variable and domain in an indexed intersection, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷
Theorem	cbvmptvw2 36854*	Change bound variable and domain in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	cbvdisjvw2 36855*	Change bound variable and domain in a disjoint collection, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷)
Theorem	cbvriotavw2 36856*	Change bound variable and domain in a restricted description binder, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐵 𝜓)
Theorem	cbvoprab1vw 36857*	Change the first bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑤 → (𝜓 ↔ 𝜒))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒}
Theorem	cbvoprab2vw 36858*	Change the second bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒}
Theorem	cbvoprab123vw 36859*	Change all bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒}
Theorem	cbvoprab23vw 36860*	Change the second and third bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒}
Theorem	cbvoprab13vw 36861*	Change the first and third bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒}
Theorem	cbvmpovw2 36862*	Change bound variables and domains in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹)
Theorem	cbvmpo1vw2 36863*	Change domains and the first bound variable in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑧 → 𝐸 = 𝐹)    &   ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑧 → 𝐴 = 𝐵)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹)
Theorem	cbvmpo2vw2 36864*	Change domains and the second bound variable in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑦 = 𝑧 → 𝐸 = 𝐹)    &   ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)    &   ⊢ (𝑦 = 𝑧 → 𝐴 = 𝐵)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑧 ∈ 𝐷 ↦ 𝐹)
Theorem	cbvixpvw2 36865*	Change bound variable and domain in an indexed Cartesian product, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷
Theorem	cbvsumvw2 36866*	Change bound variable and the set of integers in a sum, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑗 = 𝑘 → 𝐶 = 𝐷)    ⇒   ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷
Theorem	cbvprodvw2 36867*	Change bound variable and the set of integers in a product, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑗 = 𝑘 → 𝐶 = 𝐷)    ⇒   ⊢ ∏𝑗 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷
Theorem	cbvitgvw2 36868*	Change bound variable and domain in an integral, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦
Theorem	cbvditgvw2 36869*	Change bound variable and domain in a directed integral, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    &   ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)    ⇒   ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦
21.12.2.2  Change bound variables, deduction versions
Theorem	cbvmodavw 36870*	Change bound variable in the at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒))
Theorem	cbveudavw 36871*	Change bound variable in the existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))
Theorem	cbvrmodavw 36872*	Change bound variable in the restricted at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐴 𝜒))
Theorem	cbvreudavw 36873*	Change bound variable in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))
Theorem	cbvsbdavw 36874*	Change bound variable in proper substitution. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑧 / 𝑥]𝜓 ↔ [𝑧 / 𝑦]𝜒))
Theorem	cbvsbdavw2 36875*	Change bound variable in proper substitution. General version of cbvsbdavw 36874. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (𝜑 → 𝑧 = 𝑤)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑧 / 𝑥]𝜓 ↔ [𝑤 / 𝑦]𝜒))
Theorem	cbvabdavw 36876*	Change bound variable in class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
Theorem	cbvsbcdavw 36877*	Change bound variable of a class substitution. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑦]𝜒))
Theorem	cbvsbcdavw2 36878*	Change bound variable of a class substitution. General version of cbvsbcdavw 36877. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑦]𝜒))
Theorem	cbvcsbdavw 36879*	Change bound variable of a proper substitution into a class. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)
Theorem	cbvcsbdavw2 36880*	Change bound variable of a proper substitution into a class. General version of cbvcsbdavw 36879. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷)
Theorem	cbvrabdavw 36881*	Change bound variable in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∈ 𝐴 ∣ 𝜒})
Theorem	cbviundavw 36882*	Change bound variable in indexed unions. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶)
Theorem	cbviindavw 36883*	Change bound variable in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶)
Theorem	cbvopab1davw 36884*	Change the first bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑦⟩ ∣ 𝜒})
Theorem	cbvopab2davw 36885*	Change the second bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑦 = 𝑧) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑧⟩ ∣ 𝜒})
Theorem	cbvopabdavw 36886*	Change bound variables in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑤⟩ ∣ 𝜒})
Theorem	cbvmptdavw 36887*	Change bound variable in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶))
Theorem	cbvdisjdavw 36888*	Change bound variable in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶))
Theorem	cbviotadavw 36889*	Change bound variable in a description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))
Theorem	cbvriotadavw 36890*	Change bound variable in a restricted description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑦 ∈ 𝐴 𝜒))
Theorem	cbvoprab1davw 36891*	Change the first bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Theorem	cbvoprab2davw 36892*	Change the second bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒})
Theorem	cbvoprab3davw 36893*	Change the third bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑧 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜒})
Theorem	cbvoprab123davw 36894*	Change all bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒})
Theorem	cbvoprab12davw 36895*	Change the first and second bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜒})
Theorem	cbvoprab23davw 36896*	Change the second and third bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒})
Theorem	cbvoprab13davw 36897*	Change the first and third bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒})
Theorem	cbvixpdavw 36898*	Change bound variable in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶)
Theorem	cbvsumdavw 36899*	Change bound variable in a sum. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑘 = 𝑗) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑗 ∈ 𝐴 𝐶)
Theorem	cbvproddavw 36900*	Change bound variable in a product. Deduction form. (Contributed by GG, 14-Aug-2025.)
⊢ ((𝜑 ∧ 𝑗 = 𝑘) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)