P. 397 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39601-39700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfpeters2 39601*	Alternate definition of PetErs in fully modular form. This expands the Ers 𝑛 predicate into: (i) a typedness module ( Rels × CoMembErs ), (ii) an equivalence module for the coset relation ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels, (iii) the corresponding quotient-carrier (domain quotient) equation dom ≀ (...) / ≀ (...) = 𝑛. This is the equivalence-side counterpart of the modular decomposition dfpetparts2 39599 on the partition side. (Contributed by Peter Mazsa, 25-Feb-2026.)
⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})
Theorem	typesafepets 39602	Type-safe pets 39593 scheme. On a membership block-carrier 𝐴 ∈ MembParts, the lifted span (𝑅 ⋉ (◡ E ↾ 𝐴)) yields a generalized partition of 𝐴 iff its coset relation yields an equivalence relation on the same carrier 𝐴. This is the type-safe replacement for the earlier broad pets 39593: it explicitly restricts to carriers where 𝐴 is already known to be a block-family (by MembParts). That removes the standard type-safety objection ("are you equating a quotient-carrier of blocks with raw witnesses?") by construction. It is the key bridge used to identify the partition-side and equivalence-side pet classes (petseq 39603), in complete parallel with the membership bridge mpets 39583. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39603 after typedness is enforced by the "Pet*" definitions. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))
Theorem	petseq 39603	Generalized partition-equivalence identification. The partition-side scheme PetParts and the equivalence-side scheme PetErs define the same class of spans (pairs ⟨𝑟, 𝑛⟩). This plays the same organizational role for lifted spans that mpets 39583 plays for carriers: mpets 39583 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 39603 identifies the corresponding span-level predicates built from Parts and Ers. Unlike the earlier broad pets 39593, the bridge used here is the type-safe span theorem typesafepets 39602, which restricts to membership block-carriers. Since typedness (𝑟 ∈ Rels and the appropriate carrier condition) is now built directly into PetParts and PetErs, this theorem can be used downstream without repeatedly re-establishing basic typing premises. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ PetParts = PetErs
Theorem	pets2eq 39604	Grade-stable generalized partition-equivalence identification. After applying the same grade-stability operator (SucMap ShiftStable) to both sides, the grade-stable pet classes still coincide. Confirms that the grade/tower infrastructure is orthogonal to the partition-vs-equivalence viewpoint: stability is preserved under the PetParts = PetErs identification. This is the level at which we can freely work on whichever side is more convenient (Parts for block discipline, Ers for equivalence reasoning), without changing the stable notion of "pet". (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Parts = Pet2Ers
21.27  Mathbox for Rodolfo Medina
21.27.1  Partitions
Theorem	prtlem60 39605	Lemma for prter3 39634. (Contributed by Rodolfo Medina, 9-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
Theorem	bicomdd 39606	Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜒)))
Theorem	jca2r 39607	Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜓 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ∧ 𝜒)))
Theorem	jca3 39608	Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏))))
Theorem	prtlem70 39609	Lemma for prter3 39634: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
⊢ ((((𝜓 ∧ 𝜂) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃 ∧ 𝜏)))) ∧ 𝜂))
Theorem	ibdr 39610	Reverse of ibd 272. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	prtlem100 39611	Lemma for prter3 39634. (Contributed by Rodolfo Medina, 19-Oct-2010.)
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑))
Theorem	prtlem5 39612*	Lemma for prter1 39631, prter2 39633, prter3 39634 and prtex 39632. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))
Theorem	prtlem80 39613	Lemma for prter2 39633. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
Theorem	brabsb2 39614*	A closed form of brabsb 5517. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	eqbrrdv2 39615*	Other version of eqbrrdiv 5782. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	prtlem9 39616*	Lemma for prter3 39634. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
Theorem	prtlem10 39617*	Lemma for prter3 39634. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))
Theorem	prtlem11 39618	Lemma for prter2 39633. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))
Theorem	prtlem12 39619*	Lemma for prtex 39632 and prter3 39634. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )
Theorem	prtlem13 39620*	Lemma for prter1 39631, prter2 39633, prter3 39634 and prtex 39632. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
Theorem	prtlem16 39621*	Lemma for prtex 39632, prter2 39633 and prter3 39634. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ dom ∼ = ∪ 𝐴
Theorem	prtlem400 39622*	Lemma for prter2 39633 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )
Syntax	wprt 39623	Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴
Definition	df-prt 39624*	Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
Theorem	erprt 39625	The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))
Theorem	prtlem14 39626*	Lemma for prter1 39631, prter2 39633 and prtex 39632. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))
Theorem	prtlem15 39627*	Lemma for prter1 39631 and prtex 39632. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)))
Theorem	prtlem17 39628*	Lemma for prter2 39633. (Contributed by Rodolfo Medina, 15-Oct-2010.)
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))
Theorem	prtlem18 39629*	Lemma for prter2 39633. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤)))
Theorem	prtlem19 39630*	Lemma for prter2 39633. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))
Theorem	prter1 39631*	Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)
Theorem	prtex 39632*	The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V))
Theorem	prter2 39633*	The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅}))
Theorem	prter3 39634*	For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆)
21.28  Mathbox for Norm Megill We are sad to report the passing of Metamath creator and long-time contributor Norm Megill (1950 - 2021). Norm of course was the author of the Metamath proof language, the specification, all of the early tools (and some of the later ones), and the foundational work in logic and set theory for set.mm. His tools, now at https://github.com/metamath/metamath-exe, include a proof verifier, a proof assistant, a proof minimizer, style checking and reformatting, and tools for searching and displaying proofs. One of his key insights was that formal proofs can exist not only to be verified by computers, but also to be read by humans. Both the specification of the proof format (which stores full proofs, as opposed to the proof templates used by most proof assistants) and the generated web display of Metamath proofs, one of its distinctive features, contribute to this double objective. Metamath innovated both by using a very simple substitution rule (and then using that to build more complicated notions like free and bound variables) and also by taking the axiom schemas found in many theories and taking them to the next level - by making all axioms, theorems and proofs operate in terms of schemas. Not content to create Metamath for his own amusement, he also published it for the world and encouraged the development of a community of people who contributed to it and created their own tools. He was an active participant in the Metamath mailing list and other forums until days before his passing. It is often our custom to supply a quote from someone memorialized in a mathbox entry. And it is difficult to select a quote for someone who has written so much about Metamath over the years. But here is one quote from the Metamath web page which illustrates not just his clear thinking about what Metamath can and cannot do but also his desire to encourage students at all levels: Q: Will Metamath help me learn abstract mathematics? A: Yes, but probably not by itself. In order to follow a proof in an advanced math textbook, you may need to know prerequisites that could take years to learn. Some people find this frustrating. In contrast, Metamath uses a single, simple substitution rule that allows you to follow any proof mechanically. You can actually jump in anywhere and be convinced that the symbol string you see in a proof step is a consequence of the symbol strings in the earlier steps that it references, even if you don't understand what the symbols mean. But this is quite different from understanding the meaning of the math that results. Metamath alone probably will not give you an intuitive feel for abstract math, in the same way it can be hard to grasp a large computer program just by reading its source code, even though you may understand each individual instruction. However, the Bibliographic Cross-Reference lets you compare informal proofs in math textbooks and see all the implicit missing details "left to the reader."
21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16 These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems axc4 , sp 2219, axc7 2350, axc10 2417, axc11 2462, axc11n 2458, axc15 2454, axc9 2414, axc14 2495, and axc16 2297.
Axiom	ax-c5 39635	Axiom of Specialization. A universally quantified wff implies the wff without the universal quantifier (i.e., an instance, or special case, of the generalized wff). In other words, if something is true for all 𝑥, then it is true for any specific 𝑥 (that would typically occur as a free variable in the wff substituted for 𝜑). (A free variable is one that does not occur in the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦, but only 𝑥 is free in ∀𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1825. Conditional forms of the converse are given by ax-13 2404, ax-c14 39643, ax-c16 39644, and ax-5 1940. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. In our axiomatization, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution (see stdpc4 2102). An interesting alternate axiomatization uses axc5c711 39670 and ax-c4 39636 in place of ax-c5 39635, ax-4 1839, ax-10 2176, and ax-11 2192. This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2219. (Contributed by NM, 3-Jan-1993.) Use sp 2219 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Axiom	ax-c4 39636	Axiom of Quantified Implication. This axiom moves a universal quantifier from outside to inside an implication, quantifying 𝜓. Notice that 𝑥 must not be a free variable in the antecedent of the quantified implication, and we express this by binding 𝜑 to "protect" the axiom from a 𝜑 containing a free 𝑥. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc4 2354. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Axiom	ax-c7 39637	Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use axc5c711 39670 in place of ax-c5 39635, ax-c7 39637, and ax-11 2192. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc7 2350. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Axiom	ax-c10 39638	A variant of ax6 2416. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint). This axiom is obsolete and should no longer be used. It is proved above as Theorem axc10 2417. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Axiom	ax-c11 39639	Axiom ax-c11 39639 was the original version of ax-c11n 39640 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 39640 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint). This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11 2462. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Axiom	ax-c11n 39640	Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). The original version of this axiom was ax-c11 39639 and was replaced with this shorter ax-c11n 39640 ("n" for "new") in May 2008. The old axiom is proved from this one as Theorem axc11 2462. Conversely, this axiom is proved from ax-c11 39639 as Theorem axc11nfromc11 39678. This axiom was proved redundant in July 2015. See Theorem axc11n 2458. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11n 2458. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Axiom	ax-c15 39641	Axiom ax-c15 39641 was the original version of ax-12 2213, before it was discovered (in Jan. 2007) that the shorter ax-12 2213 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦 →..." as informally meaning "if 𝑥 and 𝑦 are distinct variables then..." The antecedent becomes false if the same variable is substituted for 𝑥 and 𝑦, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor". Interestingly, if the wff expression substituted for 𝜑 contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of Axiom ax-c15 39641 (from which the ax-12 2213 instance follows by Theorem ax12 2455.) The proof is by induction on formula length, using ax12eq 39693 and ax12el 39694 for the basis steps and ax12indn 39695, ax12indi 39696, and ax12inda 39700 for the induction steps. (This paragraph is true provided we use ax-c11 39639 in place of ax-c11n 39640.) This axiom is obsolete and should no longer be used. It is proved above as Theorem axc15 2454, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Axiom	ax-c9 39642	Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc9 2414. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Axiom	ax-c14 39643	Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-5 1940; see Theorem axc14 2495. Alternately, ax-5 1940 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1940. We retain ax-c14 39643 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1940, which might be easier to study for some theoretical purposes. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc14 2495. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
Axiom	ax-c16 39644*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1940 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 5420), but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-5 1940; see Theorem axc16 2297. Alternately, ax-5 1940 becomes logically redundant in the presence of this axiom, but without ax-5 1940 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 39644 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1940, which might be easier to study for some theoretical purposes. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc16 2297. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old Theorems ax12fromc15 39657 and ax13fromc9 39658 require some intermediate theorems that are included in this section.
Theorem	axc5 39645	This theorem repeats sp 2219 under the name axc5 39645, so that the Metamath program "MM> VERIFY MARKUP" command will check that it matches axiom scheme ax-c5 39635. (Contributed by NM, 18-Aug-2017.) (Proof modification is discouraged.) Use sp 2219 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	ax4fromc4 39646	Rederivation of Axiom ax-4 1839 from ax-c4 39636, ax-c5 39635, ax-gen 1825 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2354 for the derivation of ax-c4 39636 from ax-4 1839. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1839 instead. (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	ax10fromc7 39647	Rederivation of Axiom ax-10 2176 from ax-c7 39637, ax-c4 39636, ax-c5 39635, ax-gen 1825 and propositional calculus. See axc7 2350 for the derivation of ax-c7 39637 from ax-10 2176. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2176 instead. (New usage is discouraged.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	ax6fromc10 39648	Rederivation of Axiom ax-6 1997 from ax-c7 39637, ax-c10 39638, ax-gen 1825 and propositional calculus. See axc10 2417 for the derivation of ax-c10 39638 from ax-6 1997. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 1997 instead. (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	hba1-o 39649	The setvar 𝑥 is not free in ∀𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	axc4i-o 39650	Inference version of ax-c4 39636. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	equid1 39651	Proof of equid 2042 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1940; see the proof of equid 2042. See equid1ALT 39677 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	equcomi1 39652	Proof of equcomi 2047 from equid1 39651, avoiding use of ax-5 1940 (the only use of ax-5 1940 is via ax7 2046, so using ax-7 2038 instead would remove dependency on ax-5 1940). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	aecom-o 39653	Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2459 using ax-c11 39639. Unlike axc11nfromc11 39678, this version does not require ax-5 1940 (see comment of equcomi1 39652). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	aecoms-o 39654	A commutation rule for identical variable specifiers. Version of aecoms 2460 using ax-c11 39639. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	hbae-o 39655	All variables are effectively bound in an identical variable specifier. Version of hbae 2463 using ax-c11 39639. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	dral1-o 39656	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2471 using ax-c11 39639. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	ax12fromc15 39657	Rederivation of Axiom ax-12 2213 from ax-c15 39641, ax-c11 39639 (used through dral1-o 39656), and other older axioms. See Theorem axc15 2454 for the derivation of ax-c15 39641 from ax-12 2213. An open problem is whether we can prove this using ax-c11n 39640 instead of ax-c11 39639. This proof uses newer axioms ax-4 1839 and ax-6 1997, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 39636 and ax-c10 39638. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax13fromc9 39658	Derive ax-13 2404 from ax-c9 39642 and other older axioms. This proof uses newer axioms ax-4 1839 and ax-6 1997, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 39636 and ax-c10 39638. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
21.28.3  Legacy theorems using obsolete axioms These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.
Theorem	ax5ALT 39659*	Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113. (This theorem simply repeats ax-5 1940 so that we can include the following note, which applies only to the obsolete axiomatization.) This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1825, ax-c4 39636, ax-c5 39635, ax-11 2192, ax-c7 39637, ax-7 2038, ax-c9 39642, ax-c10 39638, ax-c11 39639, ax-8 2145, ax-9 2153, ax-c14 39643, ax-c15 39641, and ax-c16 39644: in that system, we can derive any instance of ax-5 1940 not containing wff variables by induction on formula length, using ax5eq 39684 and ax5el 39689 for the basis together with hbn 2330, hbal 2202, and hbim 2334. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)
Theorem	sps-o 39660	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	hbequid 39661	Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 39638.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
Theorem	nfequid-o 39662	Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1839, ax-7 2038, ax-c9 39642, and ax-gen 1825. This shows that this can be proved without ax6 2416, even though Theorem equid 2042 cannot. A shorter proof using ax6 2416 is obtainable from equid 2042 and hbth 1833.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1998, which is used for the derivation of axc9 2414, unless we consider ax-c9 39642 the starting axiom rather than ax-13 2404. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑦 𝑥 = 𝑥
Theorem	axc5c7 39663	Proof of a single axiom that can replace ax-c5 39635 and ax-c7 39637. See axc5c7toc5 39664 and axc5c7toc7 39665 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)
Theorem	axc5c7toc5 39664	Rederivation of ax-c5 39635 from axc5c7 39663. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axc5c7toc7 39665	Rederivation of ax-c7 39637 from axc5c7 39663. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc711 39666	Proof of a single axiom that can replace both ax-c7 39637 and ax-11 2192. See axc711toc7 39668 and axc711to11 39669 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑦∀𝑥𝜑 → ∀𝑦𝜑)
Theorem	nfa1-o 39667	𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥∀𝑥𝜑
Theorem	axc711toc7 39668	Rederivation of ax-c7 39637 from axc711 39666. Note that ax-c7 39637 and ax-11 2192 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc711to11 39669	Rederivation of ax-11 2192 from axc711 39666. Note that ax-c7 39637 and ax-11 2192 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	axc5c711 39670	Proof of a single axiom that can replace ax-c5 39635, ax-c7 39637, and ax-11 2192 in a subsystem that includes these axioms plus ax-c4 39636 and ax-gen 1825 (and propositional calculus). See axc5c711toc5 39671, axc5c711toc7 39672, and axc5c711to11 39673 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 39663. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Theorem	axc5c711toc5 39671	Rederivation of ax-c5 39635 from axc5c711 39670. Only propositional calculus is used by the rederivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axc5c711toc7 39672	Rederivation of ax-c7 39637 from axc5c711 39670. Note that ax-c7 39637 and ax-11 2192 are not used by the rederivation. The use of alimi 1841 (which uses ax-c5 39635) is allowed since we have already proved axc5c711toc5 39671. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc5c711to11 39673	Rederivation of ax-11 2192 from axc5c711 39670. Note that ax-c7 39637 and ax-11 2192 are not used by the rederivation. The use of alimi 1841 (which uses ax-c5 39635) is allowed since we have already proved axc5c711toc5 39671. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	equidqe 39674	equid 2042 with existential quantifier without using ax-c5 39635 or ax-5 1940. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
Theorem	axc5sp1 39675	A special case of ax-c5 39635 without using ax-c5 39635 or ax-5 1940. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
Theorem	equidq 39676	equid 2042 with universal quantifier without using ax-c5 39635 or ax-5 1940. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑦 𝑥 = 𝑥
Theorem	equid1ALT 39677	Alternate proof of equid 2042 and equid1 39651 from older axioms ax-c7 39637, ax-c10 39638 and ax-c9 39642. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	axc11nfromc11 39678	Rederivation of ax-c11n 39640 from original version ax-c11 39639. See Theorem axc11 2462 for the derivation of ax-c11 39639 from ax-c11n 39640. This theorem should not be referenced in any proof. Instead, use ax-c11n 39640 above so that uses of ax-c11n 39640 can be more easily identified, or use aecom-o 39653 when this form is needed for studies involving ax-c11 39639 and omitting ax-5 1940. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	naecoms-o 39679	A commutation rule for distinct variable specifiers. Version of naecoms 2461 using ax-c11 39639. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	hbnae-o 39680	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2464 using ax-c11 39639. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	dvelimf-o 39681	Proof of dvelimh 2482 that uses ax-c11 39639 but not ax-c15 39641, ax-c11n 39640, or ax-12 2213. Version of dvelimh 2482 using ax-c11 39639 instead of axc11 2462. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑧𝜓)    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	dral2-o 39682	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 2470 using ax-c11 39639. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	aev-o 39683*	A "distinctor elimination" lemma with no disjoint variable conditions on variables in the consequent, proved without using ax-c16 39644. Version of aev 2089 using ax-c11 39639. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Theorem	ax5eq 39684*	Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1940 considered as a metatheorem. Do not use it for later proofs - use ax-5 1940 instead, to avoid reference to the redundant axiom ax-c16 39644.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
Theorem	dveeq2-o 39685*	Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2410 using ax-c15 39641. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	axc16g-o 39686*	A generalization of Axiom ax-c16 39644. Version of axc16g 2296 using ax-c11 39639. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	dveeq1-o 39687*	Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2412 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	dveeq1-o16 39688*	Version of dveeq1 2412 using ax-c16 39644 instead of ax-5 1940. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1940. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax5el 39689*	Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1940 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)
Theorem	axc11n-16 39690*	This theorem shows that, given ax-c16 39644, we can derive a version of ax-c11n 39640. However, it is weaker than ax-c11n 39640 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Theorem	dveel2ALT 39691*	Alternate proof of dveel2 2494 using ax-c16 39644 instead of ax-5 1940. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
Theorem	ax12f 39692	Basis step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. We can start with any formula 𝜑 in which 𝑥 is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	ax12eq 39693	Basis step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝑧 = 𝑤))))
Theorem	ax12el 39694	Basis step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 ∈ 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑤))))
Theorem	ax12indn 39695	Induction step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))
Theorem	ax12indi 39696	Induction step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑 → 𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))))
Theorem	ax12indalem 39697	Lemma for ax12inda2 39699 and ax12inda 39700. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
Theorem	ax12inda2ALT 39698*	Alternate proof of ax12inda2 39699, slightly more direct and not requiring ax-c16 39644. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Theorem	ax12inda2 39699*	Induction step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 39700. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Theorem	ax12inda 39700*	Induction step for constructing a substitution instance of ax-c15 39641 without using ax-c15 39641. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 39699 may be used instead to avoid the dummy variable 𝑤 in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))