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Theorem List for Metamath Proof Explorer - 35001-35100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqbrrdv2 35001* Other version of eqbrrdiv 5465. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremprtlem9 35002* Lemma for prter3 35020. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )

Theoremprtlem10 35003* Lemma for prter3 35020. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))

Theoremprtlem11 35004 Lemma for prter2 35019. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))

Theoremprtlem12 35005* Lemma for prtex 35018 and prter3 35020. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )

Theoremprtlem13 35006* Lemma for prter1 35017, prter2 35019, prter3 35020 and prtex 35018. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))

Theoremprtlem16 35007* Lemma for prtex 35018, prter2 35019 and prter3 35020. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       dom = 𝐴

Theoremprtlem400 35008* Lemma for prter2 35019 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}        ¬ ∅ ∈ ( 𝐴 / )

Syntaxwprt 35009 Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴

Definitiondf-prt 35010* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))

Theoremerprt 35011 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( Er 𝑋 → Prt (𝐴 / ))

Theoremprtlem14 35012* Lemma for prter1 35017, prter2 35019 and prtex 35018. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))

Theoremprtlem15 35013* Lemma for prter1 35017 and prtex 35018. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))

Theoremprtlem17 35014* Lemma for prter2 35019. (Contributed by Rodolfo Medina, 15-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))

Theoremprtlem18 35015* Lemma for prter2 35019. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))

Theoremprtlem19 35016* Lemma for prter2 35019. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))

Theoremprter1 35017* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 Er 𝐴)

Theoremprtex 35018* 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⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))

Theoremprter2 35019* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))

Theoremprter3 35020* 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⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → = 𝑆)

20.23  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."

20.23.1  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 2166, axc7 2291, axc10 2348, axc11 2395, axc11n 2391, axc15 2386, axc9 2345, axc14 2447, and axc16 2233.

Axiomax-c5 35021 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all 𝑥, 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 1839. Conditional forms of the converse are given by ax-13 2333, ax-c14 35029, ax-c16 35030, and ax-5 1953.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 2427.

An interesting alternate axiomatization uses axc5c711 35056 and ax-c4 35022 in place of ax-c5 35021, ax-4 1853, ax-10 2134, and ax-11 2149.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2166. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥𝜑𝜑)

Axiomax-c4 35022 Axiom of Quantified Implication. This axiom moves a 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 2295. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Axiomax-c7 35023 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 35056 in place of ax-c5 35021, ax-c7 35023, and ax-11 2149.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc7 2291. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Axiomax-c10 35024 A variant of ax6 2347. 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 2348. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Axiomax-c11 35025 Axiom ax-c11 35025 was the original version of ax-c11n 35026 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 35026 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 2395. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Axiomax-c11n 35026 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 35025 and was replaced with this shorter ax-c11n 35026 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2395. Conversely, this axiom is proved from ax-c11 35025 as theorem axc11nfromc11 35064.

This axiom was proved redundant in July 2015. See theorem axc11n 2391.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2391. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Axiomax-c15 35027 Axiom ax-c15 35027 was the original version of ax-12 2162, before it was discovered (in Jan. 2007) that the shorter ax-12 2162 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 35027 (from which the ax-12 2162 instance follows by theorem ax12 2388.) The proof is by induction on formula length, using ax12eq 35079 and ax12el 35080 for the basis steps and ax12indn 35081, ax12indi 35082, and ax12inda 35086 for the induction steps. (This paragraph is true provided we use ax-c11 35025 in place of ax-c11n 35026.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2386, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Axiomax-c9 35028 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 2345. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Axiomax-c14 35029 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 1953; see theorem axc14 2447. Alternately, ax-5 1953 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1953. We retain ax-c14 35029 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1953, 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 2447. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Axiomax-c16 35030* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1953 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 5082), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-5 1953; see theorem axc16 2233. Alternately, ax-5 1953 becomes logically redundant in the presence of this axiom, but without ax-5 1953 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 35030 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1953, 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 2233. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

20.23.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old

Theorems ax12fromc15 35043 and ax13fromc9 35044 require some intermediate theorems that are included in this section.

Theoremaxc5 35031 This theorem repeats sp 2166 under the name axc5 35031, so that the Metamath program "MM> VERIFY MARKUP" command will check that it matches axiom scheme ax-c5 35021. (Contributed by NM, 18-Aug-2017.) (Proof modification is discouraged.) Use sp 2166 instead. (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremax4fromc4 35032 Rederivation of axiom ax-4 1853 from ax-c4 35022, ax-c5 35021, ax-gen 1839 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2295 for the derivation of ax-c4 35022 from ax-4 1853. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1853 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremax10fromc7 35033 Rederivation of axiom ax-10 2134 from ax-c7 35023, ax-c4 35022, ax-c5 35021, ax-gen 1839 and propositional calculus. See axc7 2291 for the derivation of ax-c7 35023 from ax-10 2134. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2134 instead. (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremax6fromc10 35034 Rederivation of axiom ax-6 2021 from ax-c7 35023, ax-c10 35024, ax-gen 1839 and propositional calculus. See axc10 2348 for the derivation of ax-c10 35024 from ax-6 2021. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 2021 instead. (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremhba1-o 35035 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 → ∀𝑥𝑥𝜑)

Theoremaxc4i-o 35036 Inference version of ax-c4 35022. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremequid1 35037 Proof of equid 2058 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 1953; see the proof of equid 2058. See equid1ALT 35063 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremequcomi1 35038 Proof of equcomi 2063 from equid1 35037, avoiding use of ax-5 1953 (the only use of ax-5 1953 is via ax7 2062, so using ax-7 2054 instead would remove dependency on ax-5 1953). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremaecom-o 35039 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 2392 using ax-c11 35025. Unlike axc11nfromc11 35064, this version does not require ax-5 1953 (see comment of equcomi1 35038). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecoms-o 35040 A commutation rule for identical variable specifiers. Version of aecoms 2393 using ax-c11 35025. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbae-o 35041 All variables are effectively bound in an identical variable specifier. Version of hbae 2396 using ax-c11 35025. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremdral1-o 35042 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 2404 using ax-c11 35025. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremax12fromc15 35043 Rederivation of axiom ax-12 2162 from ax-c15 35027, ax-c11 35025 (used through dral1-o 35042), and other older axioms. See theorem axc15 2386 for the derivation of ax-c15 35027 from ax-12 2162.

An open problem is whether we can prove this using ax-c11n 35026 instead of ax-c11 35025.

This proof uses newer axioms ax-4 1853 and ax-6 2021, 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 35022 and ax-c10 35024. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13fromc9 35044 Derive ax-13 2333 from ax-c9 35028 and other older axioms.

This proof uses newer axioms ax-4 1853 and ax-6 2021, 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 35022 and ax-c10 35024. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

20.23.3  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.

Theoremax5ALT 35045* 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 1953 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 1839, ax-c4 35022, ax-c5 35021, ax-11 2149, ax-c7 35023, ax-7 2054, ax-c9 35028, ax-c10 35024, ax-c11 35025, ax-8 2108, ax-9 2115, ax-c14 35029, ax-c15 35027, and ax-c16 35030: in that system, we can derive any instance of ax-5 1953 not containing wff variables by induction on formula length, using ax5eq 35070 and ax5el 35075 for the basis together with hbn 2269, hbal 2159, and hbim 2273. 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)

Theoremsps-o 35046 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theoremhbequid 35047 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 35024.) (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)

Theoremnfequid-o 35048 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 1853, ax-7 2054, ax-c9 35028, and ax-gen 1839. This shows that this can be proved without ax6 2347, even though the theorem equid 2058 cannot be. A shorter proof using ax6 2347 is obtainable from equid 2058 and hbth 1847.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 2022, which is used for the derivation of axc9 2345, unless we consider ax-c9 35028 the starting axiom rather than ax-13 2333. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremaxc5c7 35049 Proof of a single axiom that can replace ax-c5 35021 and ax-c7 35023. See axc5c7toc5 35050 and axc5c7toc7 35051 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)

Theoremaxc5c7toc5 35050 Rederivation of ax-c5 35021 from axc5c7 35049. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c7toc7 35051 Rederivation of ax-c7 35023 from axc5c7 35049. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711 35052 Proof of a single axiom that can replace both ax-c7 35023 and ax-11 2149. See axc711toc7 35054 and axc711to11 35055 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)

Theoremnfa1-o 35053 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑

Theoremaxc711toc7 35054 Rederivation of ax-c7 35023 from axc711 35052. Note that ax-c7 35023 and ax-11 2149 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711to11 35055 Rederivation of ax-11 2149 from axc711 35052. Note that ax-c7 35023 and ax-11 2149 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremaxc5c711 35056 Proof of a single axiom that can replace ax-c5 35021, ax-c7 35023, and ax-11 2149 in a subsystem that includes these axioms plus ax-c4 35022 and ax-gen 1839 (and propositional calculus). See axc5c711toc5 35057, axc5c711toc7 35058, and axc5c711to11 35059 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 35049. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c711toc5 35057 Rederivation of ax-c5 35021 from axc5c711 35056. Only propositional calculus is used by the rederivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c711toc7 35058 Rederivation of ax-c7 35023 from axc5c711 35056. Note that ax-c7 35023 and ax-11 2149 are not used by the rederivation. The use of alimi 1855 (which uses ax-c5 35021) is allowed since we have already proved axc5c711toc5 35057. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc5c711to11 35059 Rederivation of ax-11 2149 from axc5c711 35056. Note that ax-c7 35023 and ax-11 2149 are not used by the rederivation. The use of alimi 1855 (which uses ax-c5 35021) is allowed since we have already proved axc5c711toc5 35057. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremequidqe 35060 equid 2058 with existential quantifier without using ax-c5 35021 or ax-5 1953. (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

Theoremaxc5sp1 35061 A special case of ax-c5 35021 without using ax-c5 35021 or ax-5 1953. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Theoremequidq 35062 equid 2058 with universal quantifier without using ax-c5 35021 or ax-5 1953. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremequid1ALT 35063 Alternate proof of equid 2058 and equid1 35037 from older axioms ax-c7 35023, ax-c10 35024 and ax-c9 35028. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremaxc11nfromc11 35064 Rederivation of ax-c11n 35026 from original version ax-c11 35025. See theorem axc11 2395 for the derivation of ax-c11 35025 from ax-c11n 35026.

This theorem should not be referenced in any proof. Instead, use ax-c11n 35026 above so that uses of ax-c11n 35026 can be more easily identified, or use aecom-o 35039 when this form is needed for studies involving ax-c11 35025 and omitting ax-5 1953. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremnaecoms-o 35065 A commutation rule for distinct variable specifiers. Version of naecoms 2394 using ax-c11 35025. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbnae-o 35066 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2398 using ax-c11 35025. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremdvelimf-o 35067 Proof of dvelimh 2415 that uses ax-c11 35025 but not ax-c15 35027, ax-c11n 35026, or ax-12 2162. Version of dvelimh 2415 using ax-c11 35025 instead of axc11 2395. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdral2-o 35068 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 2403 using ax-c11 35025. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremaev-o 35069* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 35030. Version of aev 2099 using ax-c11 35025. (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)

Theoremax5eq 35070* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1953 considered as a metatheorem. Do not use it for later proofs - use ax-5 1953 instead, to avoid reference to the redundant axiom ax-c16 35030.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)

Theoremdveeq2-o 35071* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2341 using ax-c15 35027. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremaxc16g-o 35072* A generalization of axiom ax-c16 35030. Version of axc16g 2232 using ax-c11 35025. (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))

Theoremdveeq1-o 35073* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2343 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))

Theoremdveeq1-o16 35074* Version of dveeq1 2343 using ax-c16 35030 instead of ax-5 1953. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1953. (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax5el 35075* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1953 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 → ∀𝑧 𝑥𝑦)

Theoremaxc11n-16 35076* This theorem shows that, given ax-c16 35030, we can derive a version of ax-c11n 35026. However, it is weaker than ax-c11n 35026 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)

Theoremdveel2ALT 35077* Alternate proof of dveel2 2446 using ax-c16 35030 instead of ax-5 1953. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremax12f 35078 Basis step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. 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 = 𝑦𝜑))))

Theoremax12eq 35079 Basis step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. 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 = 𝑦𝑧 = 𝑤))))

Theoremax12el 35080 Basis step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧𝑤 → ∀𝑥(𝑥 = 𝑦𝑧𝑤))))

Theoremax12indn 35081 Induction step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))

Theoremax12indi 35082 Induction step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))))

Theoremax12indalem 35083 Lemma for ax12inda2 35085 and ax12inda 35086. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))

Theoremax12inda2ALT 35084* Alternate proof of ax12inda2 35085, slightly more direct and not requiring ax-c16 35030. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda2 35085* Induction step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 35086. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda 35086* Induction step for constructing a substitution instance of ax-c15 35027 without using ax-c15 35027. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 35085 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 = 𝑤𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12v2-o 35087* Rederivation of ax-c15 35027 from ax12v 2163 (without using ax-c15 35027 or the full ax-12 2162). Thus, the hypothesis (ax12v 2163) provides an alternate axiom that can be used in place of ax-c15 35027. See also axc15 2386. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12a2-o 35088* Derive ax-c15 35027 from a hypothesis in the form of ax-12 2162, without using ax-12 2162 or ax-c15 35027. The hypothesis is weaker than ax-12 2162, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2162, if we also have ax-c11 35025, which this proof uses. As theorem ax12 2388 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 35026 instead of ax-c11 35025. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremaxc11-o 35089 Show that ax-c11 35025 can be derived from ax-c11n 35026 and ax-12 2162. An open problem is whether this theorem can be derived from ax-c11n 35026 and the others when ax-12 2162 is replaced with ax-c15 35027 or ax12v 2163. See theorem axc11nfromc11 35064 for the rederivation of ax-c11n 35026 from axc11 2395.

Normally, axc11 2395 should be used rather than ax-c11 35025 or axc11-o 35089, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremfsumshftd 35090* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 14916. The proof demonstrates how this can be derived starting from from fsumshft 14916. (Contributed by NM, 1-Nov-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗 = (𝑘𝐾)) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

Axiomax-riotaBAD 35091 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse 𝐴. See also comments for df-iota 6099. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 6883, CONFLICTS WITH (THE NEW) df-riota 6883 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
(𝑥𝐴 𝜑) = if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴}))

TheoremriotaclbgBAD 35092* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))

TheoremriotaclbBAD 35093* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
𝐴 ∈ V       (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)

Theoremriotasvd 35094* Deduction version of riotasv 35097. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   (𝜑𝐷𝐴)       ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))

Theoremriotasv2d 35095* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5115). Special case of riota2f 6904. (Contributed by NM, 2-Mar-2013.)
𝑦𝜑    &   (𝜑𝑦𝐹)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝑦 = 𝐸) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)    &   (𝜑𝐷𝐴)    &   (𝜑𝐸𝐵)    &   (𝜑𝜒)       ((𝜑𝐴𝑉) → 𝐷 = 𝐹)

Theoremriotasv2s 35096* The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5115) in the form of a substitution instance. Special case of riota2f 6904. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)

Theoremriotasv 35097* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5115). Special case of riota2f 6904. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐴 ∈ V    &   𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)

Theoremriotasv3d 35098* A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5115) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜃)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝐶 = 𝐷) → (𝜒𝜃))    &   (𝜑 → ((𝑦𝐵𝜓) → 𝜒))    &   (𝜑𝐷𝐴)    &   (𝜑 → ∃𝑦𝐵 𝜓)       ((𝜑𝐴𝑉) → 𝜃)

20.23.4  Experiments with weak deduction theorem

Theoremelimhyps 35099 A version of elimhyp 4369 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Theoremdedths 35100 A version of weak deduction theorem dedth 4362 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓       (𝜑𝜓)

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