P. 370 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	prtex 36901*	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 36902*	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 36903*	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)
20.24  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."
20.24.1  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 2177, axc7 2312, axc10 2386, axc11 2431, axc11n 2427, axc15 2423, axc9 2383, axc14 2464, and axc16 2254.
Axiom	ax-c5 36904	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 1798. Conditional forms of the converse are given by ax-13 2373, ax-c14 36912, ax-c16 36913, and ax-5 1914. 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 2072). An interesting alternate axiomatization uses axc5c711 36939 and ax-c4 36905 in place of ax-c5 36904, ax-4 1812, ax-10 2138, and ax-11 2155. This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2177. (Contributed by NM, 3-Jan-1993.) Use sp 2177 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Axiom	ax-c4 36905	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 2316. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Axiom	ax-c7 36906	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 36939 in place of ax-c5 36904, ax-c7 36906, and ax-11 2155. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc7 2312. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Axiom	ax-c10 36907	A variant of ax6 2385. 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 2386. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Axiom	ax-c11 36908	Axiom ax-c11 36908 was the original version of ax-c11n 36909 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 36909 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 2431. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Axiom	ax-c11n 36909	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 36908 and was replaced with this shorter ax-c11n 36909 ("n" for "new") in May 2008. The old axiom is proved from this one as Theorem axc11 2431. Conversely, this axiom is proved from ax-c11 36908 as Theorem axc11nfromc11 36947. This axiom was proved redundant in July 2015. See Theorem axc11n 2427. This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11n 2427. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Axiom	ax-c15 36910	Axiom ax-c15 36910 was the original version of ax-12 2172, before it was discovered (in Jan. 2007) that the shorter ax-12 2172 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 36910 (from which the ax-12 2172 instance follows by Theorem ax12 2424.) The proof is by induction on formula length, using ax12eq 36962 and ax12el 36963 for the basis steps and ax12indn 36964, ax12indi 36965, and ax12inda 36969 for the induction steps. (This paragraph is true provided we use ax-c11 36908 in place of ax-c11n 36909.) This axiom is obsolete and should no longer be used. It is proved above as Theorem axc15 2423, 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 36911	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 2383. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Axiom	ax-c14 36912	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 1914; see Theorem axc14 2464. Alternately, ax-5 1914 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1914. We retain ax-c14 36912 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1914, 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 2464. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
Axiom	ax-c16 36913*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1914 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 5360), but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-5 1914; see Theorem axc16 2254. Alternately, ax-5 1914 becomes logically redundant in the presence of this axiom, but without ax-5 1914 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 36913 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1914, 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 2254. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old Theorems ax12fromc15 36926 and ax13fromc9 36927 require some intermediate theorems that are included in this section.
Theorem	axc5 36914	This theorem repeats sp 2177 under the name axc5 36914, so that the Metamath program "MM> VERIFY MARKUP" command will check that it matches axiom scheme ax-c5 36904. (Contributed by NM, 18-Aug-2017.) (Proof modification is discouraged.) Use sp 2177 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	ax4fromc4 36915	Rederivation of Axiom ax-4 1812 from ax-c4 36905, ax-c5 36904, ax-gen 1798 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2316 for the derivation of ax-c4 36905 from ax-4 1812. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1812 instead. (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	ax10fromc7 36916	Rederivation of Axiom ax-10 2138 from ax-c7 36906, ax-c4 36905, ax-c5 36904, ax-gen 1798 and propositional calculus. See axc7 2312 for the derivation of ax-c7 36906 from ax-10 2138. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2138 instead. (New usage is discouraged.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	ax6fromc10 36917	Rederivation of Axiom ax-6 1972 from ax-c7 36906, ax-c10 36907, ax-gen 1798 and propositional calculus. See axc10 2386 for the derivation of ax-c10 36907 from ax-6 1972. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 1972 instead. (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	hba1-o 36918	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 36919	Inference version of ax-c4 36905. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	equid1 36920	Proof of equid 2016 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 1914; see the proof of equid 2016. See equid1ALT 36946 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	equcomi1 36921	Proof of equcomi 2021 from equid1 36920, avoiding use of ax-5 1914 (the only use of ax-5 1914 is via ax7 2020, so using ax-7 2012 instead would remove dependency on ax-5 1914). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	aecom-o 36922	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 2428 using ax-c11 36908. Unlike axc11nfromc11 36947, this version does not require ax-5 1914 (see comment of equcomi1 36921). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	aecoms-o 36923	A commutation rule for identical variable specifiers. Version of aecoms 2429 using ax-c11 36908. (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 36924	All variables are effectively bound in an identical variable specifier. Version of hbae 2432 using ax-c11 36908. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	dral1-o 36925	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 2440 using ax-c11 36908. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	ax12fromc15 36926	Rederivation of Axiom ax-12 2172 from ax-c15 36910, ax-c11 36908 (used through dral1-o 36925), and other older axioms. See Theorem axc15 2423 for the derivation of ax-c15 36910 from ax-12 2172. An open problem is whether we can prove this using ax-c11n 36909 instead of ax-c11 36908. This proof uses newer axioms ax-4 1812 and ax-6 1972, 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 36905 and ax-c10 36907. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax13fromc9 36927	Derive ax-13 2373 from ax-c9 36911 and other older axioms. This proof uses newer axioms ax-4 1812 and ax-6 1972, 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 36905 and ax-c10 36907. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
20.24.3  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 36928*	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 1914 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 1798, ax-c4 36905, ax-c5 36904, ax-11 2155, ax-c7 36906, ax-7 2012, ax-c9 36911, ax-c10 36907, ax-c11 36908, ax-8 2109, ax-9 2117, ax-c14 36912, ax-c15 36910, and ax-c16 36913: in that system, we can derive any instance of ax-5 1914 not containing wff variables by induction on formula length, using ax5eq 36953 and ax5el 36958 for the basis together with hbn 2293, hbal 2168, and hbim 2297. 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 36929	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	hbequid 36930	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 36907.) (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 36931	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 1812, ax-7 2012, ax-c9 36911, and ax-gen 1798. This shows that this can be proved without ax6 2385, even though Theorem equid 2016 cannot. A shorter proof using ax6 2385 is obtainable from equid 2016 and hbth 1806.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1973, which is used for the derivation of axc9 2383, unless we consider ax-c9 36911 the starting axiom rather than ax-13 2373. (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 36932	Proof of a single axiom that can replace ax-c5 36904 and ax-c7 36906. See axc5c7toc5 36933 and axc5c7toc7 36934 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 36933	Rederivation of ax-c5 36904 from axc5c7 36932. 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 36934	Rederivation of ax-c7 36906 from axc5c7 36932. 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 36935	Proof of a single axiom that can replace both ax-c7 36906 and ax-11 2155. See axc711toc7 36937 and axc711to11 36938 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 36936	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 36937	Rederivation of ax-c7 36906 from axc711 36935. Note that ax-c7 36906 and ax-11 2155 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 36938	Rederivation of ax-11 2155 from axc711 36935. Note that ax-c7 36906 and ax-11 2155 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 36939	Proof of a single axiom that can replace ax-c5 36904, ax-c7 36906, and ax-11 2155 in a subsystem that includes these axioms plus ax-c4 36905 and ax-gen 1798 (and propositional calculus). See axc5c711toc5 36940, axc5c711toc7 36941, and axc5c711to11 36942 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 36932. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Theorem	axc5c711toc5 36940	Rederivation of ax-c5 36904 from axc5c711 36939. 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 36941	Rederivation of ax-c7 36906 from axc5c711 36939. Note that ax-c7 36906 and ax-11 2155 are not used by the rederivation. The use of alimi 1814 (which uses ax-c5 36904) is allowed since we have already proved axc5c711toc5 36940. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc5c711to11 36942	Rederivation of ax-11 2155 from axc5c711 36939. Note that ax-c7 36906 and ax-11 2155 are not used by the rederivation. The use of alimi 1814 (which uses ax-c5 36904) is allowed since we have already proved axc5c711toc5 36940. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	equidqe 36943	equid 2016 with existential quantifier without using ax-c5 36904 or ax-5 1914. (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 36944	A special case of ax-c5 36904 without using ax-c5 36904 or ax-5 1914. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
Theorem	equidq 36945	equid 2016 with universal quantifier without using ax-c5 36904 or ax-5 1914. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑦 𝑥 = 𝑥
Theorem	equid1ALT 36946	Alternate proof of equid 2016 and equid1 36920 from older axioms ax-c7 36906, ax-c10 36907 and ax-c9 36911. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	axc11nfromc11 36947	Rederivation of ax-c11n 36909 from original version ax-c11 36908. See Theorem axc11 2431 for the derivation of ax-c11 36908 from ax-c11n 36909. This theorem should not be referenced in any proof. Instead, use ax-c11n 36909 above so that uses of ax-c11n 36909 can be more easily identified, or use aecom-o 36922 when this form is needed for studies involving ax-c11 36908 and omitting ax-5 1914. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	naecoms-o 36948	A commutation rule for distinct variable specifiers. Version of naecoms 2430 using ax-c11 36908. (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 36949	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2433 using ax-c11 36908. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	dvelimf-o 36950	Proof of dvelimh 2451 that uses ax-c11 36908 but not ax-c15 36910, ax-c11n 36909, or ax-12 2172. Version of dvelimh 2451 using ax-c11 36908 instead of axc11 2431. (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 36951	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 2439 using ax-c11 36908. (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 36952*	A "distinctor elimination" lemma with no disjoint variable conditions on variables in the consequent, proved without using ax-c16 36913. Version of aev 2061 using ax-c11 36908. (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 36953*	Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1914 considered as a metatheorem. Do not use it for later proofs - use ax-5 1914 instead, to avoid reference to the redundant axiom ax-c16 36913.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
Theorem	dveeq2-o 36954*	Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2379 using ax-c15 36910. (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 36955*	A generalization of Axiom ax-c16 36913. Version of axc16g 2253 using ax-c11 36908. (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 36956*	Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2381 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 36957*	Version of dveeq1 2381 using ax-c16 36913 instead of ax-5 1914. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1914. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax5el 36958*	Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1914 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 36959*	This theorem shows that, given ax-c16 36913, we can derive a version of ax-c11n 36909. However, it is weaker than ax-c11n 36909 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 36960*	Alternate proof of dveel2 2463 using ax-c16 36913 instead of ax-5 1914. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
Theorem	ax12f 36961	Basis step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. 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 36962	Basis step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. 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 36963	Basis step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. 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 36964	Induction step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. 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 36965	Induction step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. 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 36966	Lemma for ax12inda2 36968 and ax12inda 36969. (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 36967*	Alternate proof of ax12inda2 36968, slightly more direct and not requiring ax-c16 36913. (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 36968*	Induction step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 36969. (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 36969*	Induction step for constructing a substitution instance of ax-c15 36910 without using ax-c15 36910. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 36968 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))))
Theorem	ax12v2-o 36970*	Rederivation of ax-c15 36910 from ax12v 2173 (without using ax-c15 36910 or the full ax-12 2172). Thus, the hypothesis (ax12v 2173) provides an alternate axiom that can be used in place of ax-c15 36910. See also axc15 2423. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	ax12a2-o 36971*	Derive ax-c15 36910 from a hypothesis in the form of ax-12 2172, without using ax-12 2172 or ax-c15 36910. The hypothesis is weaker than ax-12 2172, 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 2172, if we also have ax-c11 36908, which this proof uses. As Theorem ax12 2424 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 36909 instead of ax-c11 36908. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	axc11-o 36972	Show that ax-c11 36908 can be derived from ax-c11n 36909 and ax-12 2172. An open problem is whether this theorem can be derived from ax-c11n 36909 and the others when ax-12 2172 is replaced with ax-c15 36910 or ax12v 2173. See Theorems axc11nfromc11 36947 for the rederivation of ax-c11n 36909 from axc11 2431. Normally, axc11 2431 should be used rather than ax-c11 36908 or axc11-o 36972, 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))
Theorem	fsumshftd 36973*	Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 15501. The proof demonstrates how this can be derived starting from from fsumshft 15501. (Contributed by NM, 1-Nov-2019.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑗 = (𝑘 − 𝐾)) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)
Axiom	ax-riotaBAD 36974	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 6395. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 7241, CONFLICTS WITH (THE NEW) df-riota 7241 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
Theorem	riotaclbgBAD 36975*	Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
Theorem	riotaclbBAD 36976*	Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasvd 36977*	Deduction version of riotasv 36980. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
Theorem	riotasv2d 36978*	Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5327). Special case of riota2f 7266. (Contributed by NM, 2-Mar-2013.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝐹)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)
Theorem	riotasv2s 36979*	The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5327) in the form of a substitution instance. Special case of riota2f 7266. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)
Theorem	riotasv 36980*	Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5327). Special case of riota2f 7266. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))    ⇒   ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)
Theorem	riotasv3d 36981*	A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5327) 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.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜃)    &   ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))    &   ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)
20.24.4  Experiments with weak deduction theorem
Theorem	elimhyps 36982	A version of elimhyp 4525 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
⊢ [𝐵 / 𝑥]𝜑    ⇒   ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
Theorem	dedths 36983	A version of weak deduction theorem dedth 4518 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	renegclALT 36984	Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 11293. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
Theorem	elimhyps2 36985	Generalization of elimhyps 36982 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
⊢ [𝐵 / 𝑥]𝜑    ⇒   ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑
Theorem	dedths2 36986	Generalization of dedths 36983 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)
Theorem	nfcxfrdf 36987	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfded 36988	A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3639. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2916. (Contributed by NM, 19-Nov-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶)    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐶)
Theorem	nfded2 36989	A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g., ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4822) that starts from abidnf 3639. The last is assigned to the inference form (e.g., Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4821) whose hypotheses are satisfied using nfaba1 2916. (Contributed by NM, 19-Nov-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷)    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐷)
Theorem	nfunidALT2 36990	Deduction version of nfuni 4847. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	nfunidALT 36991	Deduction version of nfuni 4847. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	nfopdALT 36992	Deduction version of bound-variable hypothesis builder nfop 4821. This shows how the deduction version of a not-free theorem such as nfop 4821 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
20.24.5  Miscellanea
Theorem	cnaddcom 36993	Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	toycom 36994*	Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on ℂ. This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
⊢ 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}    &   ⊢ + = (+g‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)
Syntax	clsa 36995	Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms
Syntax	clsh 36996	Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp
Definition	df-lsatoms 36997*	Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
Definition	df-lshyp 36998*	Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less than the full space. (Contributed by NM, 29-Jun-2014.)
⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
Theorem	lshpset 36999*	The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Theorem	islshp 37000*	The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))