P. 392 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mainer2 39101	The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 15-Oct-2021.)
⊢ (𝑅 ErALTV 𝐴 → ( CoElEqvRel 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Theorem	mainerim 39102	Every equivalence relation implies equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.)
⊢ (𝑅 ErALTV 𝐴 → CoElEqvRel 𝐴)
Theorem	petincnvepres2 39103	A partition-equivalence theorem with intersection and general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ∩ (◡ E ↾ 𝐴)) / (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) / ≀ (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴))
Theorem	petincnvepres 39104	The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 38709. Cf. pet 39106. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝑅 ∩ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ErALTV 𝐴)
Theorem	pet2 39105	Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 39106 and pets 39107) is the main result of my investigation into set theory, see the comment of pet 39106. (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
Theorem	pet 39106	Partition-Equivalence Theorem with general 𝑅 while preserving the restricted converse epsilon relation of mpet2 39095 (as opposed to petincnvepres 39104). A class is a partition by a range Cartesian product with general 𝑅 and the restricted converse element class if and only if the cosets by the range Cartesian product are in an equivalence relation on it. Cf. br1cossxrncnvepres 38711. This theorem (together with pets 39107 and pet2 39105) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 39094, mpet2 39095 and mpet3 39091 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 39095), i.e., it is not the hypothetical General Partition-Equivalence Theorem gpet ⊢ (𝑅 Part 𝐴 ↔ ≀ 𝑅 ErALTV 𝐴), but this one has a general part that mpet2 39095 lacks: 𝑅, which is sufficient for my future application of set theory, for my purpose outside of set theory. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)
Theorem	pets 39107	Partition-Equivalence Theorem with general 𝑅, with binary relations. This theorem (together with pet 39106 and pet2 39105) is the main result of my investigation into set theory, cf. the comment of pet 39106. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))
Theorem	dmqsblocks 39108*	If the pet 39106 span (𝑅 ⋉ (◡ E ↾ 𝐴)) partitions 𝐴, then every block 𝑢 ∈ 𝐴 is of the form [𝑣] for some 𝑣 that not only lies in the domain but also has at least one internal element 𝑐 and at least one 𝑅-target 𝑏 (cf. also the comments of qseq 38903). It makes explicit that pet 39106 gives active representatives for each block, without ever forcing 𝑣 = 𝑢. (Contributed by Peter Mazsa, 23-Nov-2025.)
⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
21.27  Mathbox for Rodolfo Medina
21.27.1  Partitions
Theorem	prtlem60 39109	Lemma for prter3 39138. (Contributed by Rodolfo Medina, 9-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
Theorem	bicomdd 39110	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 39111	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 39112	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 39113	Lemma for prter3 39138: 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 39114	Reverse of ibd 269. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	prtlem100 39115	Lemma for prter3 39138. (Contributed by Rodolfo Medina, 19-Oct-2010.)
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑))
Theorem	prtlem5 39116*	Lemma for prter1 39135, prter2 39137, prter3 39138 and prtex 39136. (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 39117	Lemma for prter2 39137. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
Theorem	brabsb2 39118*	A closed form of brabsb 5479. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	eqbrrdv2 39119*	Other version of eqbrrdiv 5743. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	prtlem9 39120*	Lemma for prter3 39138. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
Theorem	prtlem10 39121*	Lemma for prter3 39138. (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 39122	Lemma for prter2 39137. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))
Theorem	prtlem12 39123*	Lemma for prtex 39136 and prter3 39138. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )
Theorem	prtlem13 39124*	Lemma for prter1 39135, prter2 39137, prter3 39138 and prtex 39136. (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 39125*	Lemma for prtex 39136, prter2 39137 and prter3 39138. (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 39126*	Lemma for prter2 39137 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 39127	Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴
Definition	df-prt 39128*	Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
Theorem	erprt 39129	The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))
Theorem	prtlem14 39130*	Lemma for prter1 39135, prter2 39137 and prtex 39136. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))
Theorem	prtlem15 39131*	Lemma for prter1 39135 and prtex 39136. (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 39132*	Lemma for prter2 39137. (Contributed by Rodolfo Medina, 15-Oct-2010.)
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))
Theorem	prtlem18 39133*	Lemma for prter2 39137. (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 39134*	Lemma for prter2 39137. (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 39135*	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 39136*	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 39137*	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 39138*	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 2190, axc7 2322, axc10 2389, axc11 2434, axc11n 2430, axc15 2426, axc9 2386, axc14 2467, and axc16 2268.
Axiom	ax-c5 39139	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 1796. Conditional forms of the converse are given by ax-13 2376, ax-c14 39147, ax-c16 39148, and ax-5 1911. 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 2073). An interesting alternate axiomatization uses axc5c711 39174 and ax-c4 39140 in place of ax-c5 39139, ax-4 1810, ax-10 2146, and ax-11 2162. This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2190. (Contributed by NM, 3-Jan-1993.) Use sp 2190 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Axiom	ax-c4 39140	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 2326. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Axiom	ax-c7 39141	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 39174 in place of ax-c5 39139, ax-c7 39141, and ax-11 2162. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc7 2322. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Axiom	ax-c10 39142	A variant of ax6 2388. 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 2389. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Axiom	ax-c11 39143	Axiom ax-c11 39143 was the original version of ax-c11n 39144 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 39144 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 2434. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Axiom	ax-c11n 39144	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 39143 and was replaced with this shorter ax-c11n 39144 ("n" for "new") in May 2008. The old axiom is proved from this one as Theorem axc11 2434. Conversely, this axiom is proved from ax-c11 39143 as Theorem axc11nfromc11 39182. This axiom was proved redundant in July 2015. See Theorem axc11n 2430. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11n 2430. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Axiom	ax-c15 39145	Axiom ax-c15 39145 was the original version of ax-12 2184, before it was discovered (in Jan. 2007) that the shorter ax-12 2184 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 39145 (from which the ax-12 2184 instance follows by Theorem ax12 2427.) The proof is by induction on formula length, using ax12eq 39197 and ax12el 39198 for the basis steps and ax12indn 39199, ax12indi 39200, and ax12inda 39204 for the induction steps. (This paragraph is true provided we use ax-c11 39143 in place of ax-c11n 39144.) This axiom is obsolete and should no longer be used. It is proved above as Theorem axc15 2426, 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 39146	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 2386. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Axiom	ax-c14 39147	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 1911; see Theorem axc14 2467. Alternately, ax-5 1911 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1911. We retain ax-c14 39147 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1911, 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 2467. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
Axiom	ax-c16 39148*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1911 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 5386), but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-5 1911; see Theorem axc16 2268. Alternately, ax-5 1911 becomes logically redundant in the presence of this axiom, but without ax-5 1911 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 39148 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1911, 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 2268. (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 39161 and ax13fromc9 39162 require some intermediate theorems that are included in this section.
Theorem	axc5 39149	This theorem repeats sp 2190 under the name axc5 39149, so that the Metamath program "MM> VERIFY MARKUP" command will check that it matches axiom scheme ax-c5 39139. (Contributed by NM, 18-Aug-2017.) (Proof modification is discouraged.) Use sp 2190 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	ax4fromc4 39150	Rederivation of Axiom ax-4 1810 from ax-c4 39140, ax-c5 39139, ax-gen 1796 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2326 for the derivation of ax-c4 39140 from ax-4 1810. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1810 instead. (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	ax10fromc7 39151	Rederivation of Axiom ax-10 2146 from ax-c7 39141, ax-c4 39140, ax-c5 39139, ax-gen 1796 and propositional calculus. See axc7 2322 for the derivation of ax-c7 39141 from ax-10 2146. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2146 instead. (New usage is discouraged.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	ax6fromc10 39152	Rederivation of Axiom ax-6 1968 from ax-c7 39141, ax-c10 39142, ax-gen 1796 and propositional calculus. See axc10 2389 for the derivation of ax-c10 39142 from ax-6 1968. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 1968 instead. (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	hba1-o 39153	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 39154	Inference version of ax-c4 39140. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	equid1 39155	Proof of equid 2013 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 1911; see the proof of equid 2013. See equid1ALT 39181 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	equcomi1 39156	Proof of equcomi 2018 from equid1 39155, avoiding use of ax-5 1911 (the only use of ax-5 1911 is via ax7 2017, so using ax-7 2009 instead would remove dependency on ax-5 1911). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	aecom-o 39157	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 2431 using ax-c11 39143. Unlike axc11nfromc11 39182, this version does not require ax-5 1911 (see comment of equcomi1 39156). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	aecoms-o 39158	A commutation rule for identical variable specifiers. Version of aecoms 2432 using ax-c11 39143. (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 39159	All variables are effectively bound in an identical variable specifier. Version of hbae 2435 using ax-c11 39143. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	dral1-o 39160	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 2443 using ax-c11 39143. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	ax12fromc15 39161	Rederivation of Axiom ax-12 2184 from ax-c15 39145, ax-c11 39143 (used through dral1-o 39160), and other older axioms. See Theorem axc15 2426 for the derivation of ax-c15 39145 from ax-12 2184. An open problem is whether we can prove this using ax-c11n 39144 instead of ax-c11 39143. This proof uses newer axioms ax-4 1810 and ax-6 1968, 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 39140 and ax-c10 39142. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax13fromc9 39162	Derive ax-13 2376 from ax-c9 39146 and other older axioms. This proof uses newer axioms ax-4 1810 and ax-6 1968, 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 39140 and ax-c10 39142. (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 39163*	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 1911 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 1796, ax-c4 39140, ax-c5 39139, ax-11 2162, ax-c7 39141, ax-7 2009, ax-c9 39146, ax-c10 39142, ax-c11 39143, ax-8 2115, ax-9 2123, ax-c14 39147, ax-c15 39145, and ax-c16 39148: in that system, we can derive any instance of ax-5 1911 not containing wff variables by induction on formula length, using ax5eq 39188 and ax5el 39193 for the basis together with hbn 2301, hbal 2172, and hbim 2305. 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 39164	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	hbequid 39165	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 39142.) (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 39166	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 1810, ax-7 2009, ax-c9 39146, and ax-gen 1796. This shows that this can be proved without ax6 2388, even though Theorem equid 2013 cannot. A shorter proof using ax6 2388 is obtainable from equid 2013 and hbth 1804.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1969, which is used for the derivation of axc9 2386, unless we consider ax-c9 39146 the starting axiom rather than ax-13 2376. (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 39167	Proof of a single axiom that can replace ax-c5 39139 and ax-c7 39141. See axc5c7toc5 39168 and axc5c7toc7 39169 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 39168	Rederivation of ax-c5 39139 from axc5c7 39167. 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 39169	Rederivation of ax-c7 39141 from axc5c7 39167. 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 39170	Proof of a single axiom that can replace both ax-c7 39141 and ax-11 2162. See axc711toc7 39172 and axc711to11 39173 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 39171	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 39172	Rederivation of ax-c7 39141 from axc711 39170. Note that ax-c7 39141 and ax-11 2162 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 39173	Rederivation of ax-11 2162 from axc711 39170. Note that ax-c7 39141 and ax-11 2162 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 39174	Proof of a single axiom that can replace ax-c5 39139, ax-c7 39141, and ax-11 2162 in a subsystem that includes these axioms plus ax-c4 39140 and ax-gen 1796 (and propositional calculus). See axc5c711toc5 39175, axc5c711toc7 39176, and axc5c711to11 39177 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 39167. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Theorem	axc5c711toc5 39175	Rederivation of ax-c5 39139 from axc5c711 39174. 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 39176	Rederivation of ax-c7 39141 from axc5c711 39174. Note that ax-c7 39141 and ax-11 2162 are not used by the rederivation. The use of alimi 1812 (which uses ax-c5 39139) is allowed since we have already proved axc5c711toc5 39175. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc5c711to11 39177	Rederivation of ax-11 2162 from axc5c711 39174. Note that ax-c7 39141 and ax-11 2162 are not used by the rederivation. The use of alimi 1812 (which uses ax-c5 39139) is allowed since we have already proved axc5c711toc5 39175. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	equidqe 39178	equid 2013 with existential quantifier without using ax-c5 39139 or ax-5 1911. (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 39179	A special case of ax-c5 39139 without using ax-c5 39139 or ax-5 1911. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
Theorem	equidq 39180	equid 2013 with universal quantifier without using ax-c5 39139 or ax-5 1911. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑦 𝑥 = 𝑥
Theorem	equid1ALT 39181	Alternate proof of equid 2013 and equid1 39155 from older axioms ax-c7 39141, ax-c10 39142 and ax-c9 39146. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	axc11nfromc11 39182	Rederivation of ax-c11n 39144 from original version ax-c11 39143. See Theorem axc11 2434 for the derivation of ax-c11 39143 from ax-c11n 39144. This theorem should not be referenced in any proof. Instead, use ax-c11n 39144 above so that uses of ax-c11n 39144 can be more easily identified, or use aecom-o 39157 when this form is needed for studies involving ax-c11 39143 and omitting ax-5 1911. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	naecoms-o 39183	A commutation rule for distinct variable specifiers. Version of naecoms 2433 using ax-c11 39143. (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 39184	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2436 using ax-c11 39143. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	dvelimf-o 39185	Proof of dvelimh 2454 that uses ax-c11 39143 but not ax-c15 39145, ax-c11n 39144, or ax-12 2184. Version of dvelimh 2454 using ax-c11 39143 instead of axc11 2434. (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 39186	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 2442 using ax-c11 39143. (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 39187*	A "distinctor elimination" lemma with no disjoint variable conditions on variables in the consequent, proved without using ax-c16 39148. Version of aev 2060 using ax-c11 39143. (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 39188*	Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1911 considered as a metatheorem. Do not use it for later proofs - use ax-5 1911 instead, to avoid reference to the redundant axiom ax-c16 39148.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
Theorem	dveeq2-o 39189*	Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2382 using ax-c15 39145. (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 39190*	A generalization of Axiom ax-c16 39148. Version of axc16g 2267 using ax-c11 39143. (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 39191*	Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2384 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 39192*	Version of dveeq1 2384 using ax-c16 39148 instead of ax-5 1911. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1911. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax5el 39193*	Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1911 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 39194*	This theorem shows that, given ax-c16 39148, we can derive a version of ax-c11n 39144. However, it is weaker than ax-c11n 39144 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 39195*	Alternate proof of dveel2 2466 using ax-c16 39148 instead of ax-5 1911. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
Theorem	ax12f 39196	Basis step for constructing a substitution instance of ax-c15 39145 without using ax-c15 39145. 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 39197	Basis step for constructing a substitution instance of ax-c15 39145 without using ax-c15 39145. 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 39198	Basis step for constructing a substitution instance of ax-c15 39145 without using ax-c15 39145. 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 39199	Induction step for constructing a substitution instance of ax-c15 39145 without using ax-c15 39145. 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 39200	Induction step for constructing a substitution instance of ax-c15 39145 without using ax-c15 39145. 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)))))