P. 370 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)

Type	Label	Description
Statement
21.11.19  Rank theorems
Theorem	rank0 36901	The rank of the empty set is ∅. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ (rank‘∅) = ∅
Theorem	rankeq1o 36902	The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})
21.11.20  Hereditarily Finite Sets
Theorem	hftr 36903	The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ Tr HF
Theorem	hfext 36904*	Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ HF (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
Theorem	hfninf 36905	ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ¬ ω ∈ HF
21.11.21  Natural ordinal operations
Syntax	cnmul 36906	Declare the syntax for natural multiplication.
class ·no
Definition	df-nmul 36907*	Define natural ordinal multiplication. This is the corresponding operation to df-nadd 8659. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ·no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑝 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑎⦌⦋(2nd ‘𝑝) / 𝑏⦌∩ {𝑧 ∈ On ∣ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 ((𝑐𝑚𝑏) +no (𝑎𝑚𝑑)) ∈ (𝑧 +no (𝑐𝑚𝑑))}))
Theorem	nmulfn 36908	Natural multiplication is a function over pairs of ordinals. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ·no Fn (On × On)
Theorem	nmulprop 36909*	Show closure and value of natural multiplication. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·no 𝐵) ∈ On ∧ (𝐴 ·no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝑥 +no (𝑎 ·no 𝑏))}))
Theorem	nmulcl 36910	Closure law for natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
Theorem	nmulval 36911*	Show the value of natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝑥 +no (𝑎 ·no 𝑏))})
Theorem	nmulcld 36912	Closure law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 12-Jun-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 ·no 𝐵) ∈ On)
Theorem	nmulcom 36913	Natural multiplication is commutative. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))
Theorem	nmulr0 36914	Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ (𝐴 ∈ On → (𝐴 ·no ∅) = ∅)
Theorem	nmull0 36915	Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ (𝐴 ∈ On → (∅ ·no 𝐴) = ∅)
Theorem	nmulrid 36916	Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ (𝐴 ∈ On → (𝐴 ·no 1o) = 𝐴)
Theorem	nmullid 36917	Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ (𝐴 ∈ On → (1o ·no 𝐴) = 𝐴)
Theorem	onelond 36918	An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. Lemma 1.3 of [Schloeder] p. 1. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ On)
Theorem	ontr2d 36919	Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	onelssd 36920	An element of an ordinal number is a subset of the number. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
Theorem	nmulr0d 36921	Natural multiplication by zero. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 ·no ∅) = ∅)
Theorem	nmull0d 36922	Natural multiplication by zero. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (∅ ·no 𝐴) = ∅)
Theorem	nmulridd 36923	Identity law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 ·no 1o) = 𝐴)
Theorem	nmullidd 36924	Identity law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (1o ·no 𝐴) = 𝐴)
Theorem	nmulcomd 36925	Natural multiplication commutes. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))
Theorem	naddridd 36926	Identity law for natural addition. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 +no ∅) = 𝐴)
Theorem	naddlidd 36927	Identity law for natural addition. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (∅ +no 𝐴) = 𝐴)
Theorem	naddcomd 36928	Natural addition commutes. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
Theorem	naddassd 36929	Natural addition associates. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    ⇒   ⊢ (𝜑 → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))
Theorem	nadd32d 36930	Commutative/associative law that swaps the last two terms in a triple sum. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    ⇒   ⊢ (𝜑 → ((𝐴 +no 𝐵) +no 𝐶) = ((𝐴 +no 𝐶) +no 𝐵))
Theorem	nmuladdel 36931	Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
Theorem	nmuladdss 36932	Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
Theorem	nmulss1 36933	Natural multiplication preserves less-than or equal. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ·no 𝐴) ⊆ (𝐶 ·no 𝐵))
Theorem	nmulel1 36934	Natural multiplication by a non-zero number preserves less-than. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ≠ ∅)) → (𝐶 ·no 𝐴) ∈ (𝐶 ·no 𝐵))
Theorem	ltnmul 36935*	Characterize less-than a natural product. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 ·no 𝐶) ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 (𝐴 +no (𝑏 ·no 𝑐)) ⊆ ((𝑏 ·no 𝐶) +no (𝐵 ·no 𝑐))))
Theorem	nmulle 36936*	A condition for bounding a natural product above. Converse of ltnmul 36935. (Contributed by Scott Fenton, 16-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·no 𝐵) ⊆ 𝐶 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝐶 +no (𝑎 ·no 𝑏))))
Theorem	ltnadd 36937*	Condition for bounding a natural sum below. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 +no 𝐶) ↔ (∃𝑏 ∈ 𝐵 𝐴 ⊆ (𝑏 +no 𝐶) ∨ ∃𝑐 ∈ 𝐶 𝐴 ⊆ (𝐵 +no 𝑐))))
Theorem	naddle 36938*	Condition for bounding natural addition above. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))
Theorem	nadddilem1 36939*	Lemma for nadddi 36943. Prove a subcase of 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 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))    &   ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
Theorem	nadddilem2 36940*	Lemma for nadddi 36943. 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 36941*	Lemma for nadddi 36943. 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 36942*	Lemma for nadddi 36943. 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 36943	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 36944	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 36945	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 36946	Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)
Theorem	rmoeqbii 36947	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 36948	Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)
Theorem	reueqbii 36949	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 36950	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 36951	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 36952	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 36953	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 36954	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 36955	Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶
Theorem	iineq12i 36956	Equality theorem for indexed intersection. Inference version. General version of iineq1i 36955. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷
Theorem	riotaeqbii 36957	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 36958	Equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑)
Theorem	ixpeq1i 36959	Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶
Theorem	ixpeq12i 36960	Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷
Theorem	sumeq2si 36961	Equality inference for sum. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	sumeq12si 36962	Equality inference for sum. General version of sumeq2si 36961. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ Σ𝑥 ∈ 𝐴 𝐶 = Σ𝑥 ∈ 𝐵 𝐷
Theorem	prodeq2si 36963	Equality inference for product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶
Theorem	prodeq12si 36964	Equality inference for product. General version of prodeq2si 36963. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
Theorem	itgeq12i 36965	Equality inference for an integral. General version of itgeq1i 36966 and itgeq2i 36967. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥
Theorem	itgeq1i 36966	Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥
Theorem	itgeq2i 36967	Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥
Theorem	ditgeq123i 36968	Equality inference for the directed integral. General version of ditgeq12i 36969 and ditgeq3i 36970. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    &   ⊢ 𝐸 = 𝐹    ⇒   ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑥
Theorem	ditgeq12i 36969	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 36970	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 36971*	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 36972*	Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3381. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒))
Theorem	sbequbidv 36973*	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 36974*	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 36975*	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 36976*	Equality deduction for sum. General version of sumeq2sdv 15850. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)
Theorem	prodeq12sdv 36977*	Equality deduction for product. General version of prodeq2sdv 16071. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷)
Theorem	itgeq12sdv 36978*	Equality theorem for an integral. Deduction form. General version of itgeq1d 46911 and itgeq2sdv 36979. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
Theorem	itgeq2sdv 36979*	Equality theorem for an integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
Theorem	ditgeq123dv 36980*	Equality theorem for the directed integral. Deduction form. General version of ditgeq3sdv 36982. (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 36981*	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 36982*	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 36983	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 36984	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 36983, this proof does not rely on df-cleq 2753, 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 36985*	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 36986*	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 36987*	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 36988*	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 36989*	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 36990*	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 36991*	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 36992*	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 36993*	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 36994*	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 36995*	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 36996*	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 36997*	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 36998*	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 36999*	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 37000*	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}