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Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwl-moae 33901* Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 2021 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 33902 and exists1 2692. Gerard Lang pointed out, that 𝑦𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2551, trut 1608) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant . (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.)
(∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theoremwl-euae 33902* Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theoremwl-nax6im 33903* The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like 𝑥 has no instance to assign to. An expression like 𝑥 = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 2005 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 881, for example. Whatever it is, we start out with the contraposition of ax-6 2021, that guarantees the existence of at least one set. Our hypothesis here expresses tentatively it might not hold. We can simplify the antecedent then, to the point where we do not need equation any more. This suggests what a decent characterization of the empty domain could be. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥 𝑥 = 𝑦𝜑)       (¬ ∃𝑥⊤ → 𝜑)

Theoremwl-nax6al 33904 In an empty domain the for-all operator always holds, even when applied to a false expression. This theorem actually shows that ax-5 1953 is provable there. Also we cannot assume that sp 2167 generally holds, except of course in the form of sptruw 1850. Without the support of an sp 2167 like theorem it seems difficult, if not impossible, to arrive at a theorem allowing to change the bounded variable in the antecedent. Additional axioms need to be postulated to further strengthen this result.

A consequence of this result is that 𝑥𝜑 is not true for any 𝜑. In particular, ∃*𝑥𝜑 does not hold either, a somewhat counterintuitive result. (Contributed by Wolf Lammen, 12-Mar-2023.)

(¬ ∃𝑥⊤ → ∀𝑥𝜑)

Theoremwl-nax6nfr 33905 All expressions are free of the variable used in the antecedent. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)

Theoremwl-naev 33906* If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Theoremwl-hbae1 33907 This specialization of hbae 2397 does not depend on ax-11 2150. (Contributed by Wolf Lammen, 8-Aug-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)

Theoremwl-naevhba1v 33908* An instance of hbn1w 2091 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-spae 33909 Prove an instance of sp 2167 from ax-13 2334 and Tarski's FOL only, without distinct variable conditions. The antecedent 𝑥𝑥 = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for 𝑥, or vice versa, i.e. both variables are effectively the same. The converse ¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies 𝑥𝑥 = 𝑦 and ¬ ∀𝑥𝑥 = 𝑦 can help eliminating distinct variable conditions.

The antecedent 𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'.

Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2163.

Note that this theorem is also provable from ax-12 2163 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 2031 and spaev 2095 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremwl-speqv 33910* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2167 is provable from Tarski's FOL and ax13v 2335 only. Note that this reverts the implication in ax13lem1 2336, so in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))

Theoremwl-19.8eqv 33911* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2166 is provable from Tarski's FOL and ax13v 2335 only. Note that this reverts the implication in ax13lem2 2338, so in fact 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦))

Theoremwl-19.2reqv 33912* Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 2026 is provable from Tarski's FOL and ax13v 2335 only. Note that in conjunction with 19.2 2026 in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremwl-nfalv 33913* If 𝑥 is not present in 𝜑, it is not free in 𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.)
𝑥𝑦𝜑

Theoremwl-nfimf1 33914 An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1943 in dvelimdf 2415 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
(∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))

Theoremwl-nfnbi 33915 Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1902 or nfnd 1903. (Contributed by Wolf Lammen, 5-May-2018.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Theoremwl-nfae1 33916 Unlike nfae 2398, this specialized theorem avoids ax-11 2150. (Contributed by Wolf Lammen, 26-Jun-2019.)
𝑥𝑦 𝑦 = 𝑥

Theoremwl-nfnae1 33917 Unlike nfnae 2400, this specialized theorem avoids ax-11 2150. (Contributed by Wolf Lammen, 27-Jun-2019.)
𝑥 ¬ ∀𝑦 𝑦 = 𝑥

Theoremwl-aetr 33918 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))

Theoremwl-dral1d 33919 A version of dral1 2405 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, 𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 33930 and nf5di 2259 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))

Theoremwl-cbvalnaed 33920 wl-cbvalnae 33921 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremwl-cbvalnae 33921 A more general version of cbval 2368 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2414, nfsb2 2436 or dveeq1 2344. (Contributed by Wolf Lammen, 4-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremwl-exeq 33922 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

Theoremwl-aleq 33923 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))

Theoremwl-nfeqfb 33924 Extend nfeqf 2345 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
(Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))

Theoremwl-nfs1t 33925 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2441. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremwl-equsald 33926 Deduction version of equsal 2382. (Contributed by Wolf Lammen, 27-Jul-2019.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Theoremwl-equsal 33927 A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 33926 first, and then deriving more specialized versions wl-equsal 33927 and wl-equsal1t 33928 then is more efficient than the other way round, which is possible, too. See also equsal 2382. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremwl-equsal1t 33928 The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑.

This theorem is more fundamental than equsal 2382, spimt 2350 or sbft 2455, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)

(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Theoremwl-equsalcom 33929 This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑦 = 𝑥𝜑))

Theoremwl-equsal1i 33930 The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
(Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)    &   (𝑥 = 𝑦𝜑)       𝜑

Theoremwl-dv-sb 33931* Substitution for 𝑥 has no effect on 𝜑 not containing 𝑥. See also sbf 2456. (Contributed by Wolf Lammen, 4-Sep-2022.)
(𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremwl-sb6rft 33932 A specialization of wl-equsal1t 33928. Closed form of sb6rf 2500. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))

Theoremwl-sbrimt 33933 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2472. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

Theoremwl-sblimt 33934 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2472. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))

Theoremwl-sb8t 33935 Substitution of variable in universal quantifier. Closed form of sb8 2501. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sb8et 33936 Substitution of variable in universal quantifier. Closed form of sb8e 2502. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sbhbt 33937 Closed form of sbhb 2518. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Theoremwl-sbnf1 33938 Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2326. Note: This theorem shows that sbnf2 2326 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Theoremwl-equsb3 33939 equsb3 2510 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
(¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))

Theoremwl-equsb4 33940 Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))

Theoremwl-sb6nae 33941 Version of sb6 2250 suitable for elimination of unnecessary disjoint variable conditions. (Contributed by Wolf Lammen, 28-Jul-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-2sb6d 33942 Version of 2sb6 2253 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
(𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)    &   (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)    &   (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)       (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))

Theoremwl-sbcom2d-lem1 33943* Lemma used to prove wl-sbcom2d 33945. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))

Theoremwl-sbcom2d-lem2 33944* Lemma used to prove wl-sbcom2d 33945. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
(¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) → 𝜑)))

Theoremwl-sbcom2d 33945 Version of sbcom2 2523 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
(𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    &   (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)       (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))

Theoremwl-sbalnae 33946 A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremwl-sbal1 33947* A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 33946 now. See also sbal1 2539. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremwl-sbal2 33948* Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 33946 now. See also sbal2 2540. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremwl-lem-exsb 33949* This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-lem-nexmo 33950 This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))

Theoremwl-lem-moexsb 33951* The antecedent 𝑥(𝜑𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑.

This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)

(∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremwl-alanbii 33952 This theorem extends alanimi 1860 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
(𝜑 ↔ (𝜓𝜒))       (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))

Theoremwl-mo2df 33953 Version of mof 2578 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Theoremwl-mo2tf 33954 Closed form of mof 2578 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-eudf 33955 Version of eu6 2592 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Theoremwl-eutf 33956 Closed form of eu6 2592 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-euequf 33957 euequ 2616 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Theoremwl-mo2t 33958* Closed form of mof 2578. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-mo3t 33959* Closed form of mo3 2580. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Theoremwl-sb8eut 33960 Substitution of variable in universal quantifier. Closed form of sb8eu 2635. (Contributed by Wolf Lammen, 11-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sb8mot 33961 Substitution of variable in universal quantifier. Closed form of sb8mo 2636.

This theorem relates to wl-mo3t 33959, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2455 and sbco 2488. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 33959 in a simple fashion, unfortunately only if 𝑥 and 𝑦 are known to be distinct. To avoid any hassle with distinctors, we prefer to derive this theorem independently, ignoring the close connection between both theorems. From an educational standpoint, one would assume wl-mo3t 33959 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 11-Aug-2019.)

(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Axiomax-wl-11v 33962* Version of ax-11 2150 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2150. It will later be converted into a theorem directly based on ax-11 2150. (Contributed by Wolf Lammen, 28-Jun-2019.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremwl-ax11-lem1 33963 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))

Theoremwl-ax11-lem2 33964* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)

Theoremwl-ax11-lem3 33965* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)

Theoremwl-ax11-lem4 33966* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-ax11-lem5 33967 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Theoremwl-ax11-lem6 33968* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))

Theoremwl-ax11-lem7 33969 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))

Theoremwl-ax11-lem8 33970* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))

Theoremwl-ax11-lem9 33971 The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))

Theoremwl-ax11-lem10 33972* We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 184 can be used in conjunction with wl-ax11-lem9 33971 to eliminate the second antecedent. Missing is something along the lines of ax-6 2021, so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑦 𝑦 = 𝑢 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)))

Theoremwl-clabv 33973* Variant of df-clab 2764, where the element 𝑥 is required to be disjoint from the class it is taken from. This restriction meets similar ones found in other definitions and axioms like ax-ext 2754, df-clel 2774 and df-cleq 2770. 𝑥𝐴 with 𝐴 depending on 𝑥 can be the source of side effects, that you rather want to be aware of. So here we eliminate one possible way of letting this slip in instead.

An expression 𝑥𝐴 with 𝑥, 𝐴 not disjoint, is now only introduced either via ax-8 2109, ax-9 2116, or df-clel 2774. Theorem cleljust 2115 shows that a possible choice does not matter.

The original df-clab 2764 can be rederived, see wl-dfclab 33974. In an implementation this theorem is the only user of df-clab. (Contributed by NM, 26-May-1993.) Element and class are disjoint. (Revised by Wolf Lammen, 31-May-2023.)

(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

Theoremwl-dfclab 33974 Rederive df-clab 2764 from wl-clabv 33973. (Contributed by Wolf Lammen, 31-May-2023.)
(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

Theoremwl-clabtv 33975* Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 33976. (Contributed by Wolf Lammen, 29-May-2023.)
(𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})

Theoremwl-clabt 33976 Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 33975. (Contributed by Wolf Lammen, 29-May-2023.)
𝑥𝜑       (𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})

20.17.5  1. Restricted Quantifiers

Syntaxwl-ral 33977 Redefine the restricted universal quantifier context to avoid overloading and syntax check errors in mmj2. This operator does not require 𝑥 and 𝐵 disjoint.
wff 𝑥 in 𝐵𝜑

Syntaxwl-rex 33978 Redefine the restricted existential quantifier context to avoid overloading and syntax check errors in mmj2. This operator does not require 𝑥 and 𝐵 disjoint.
wff 𝑥 in 𝐵𝜑

Syntaxwl-rmo 33979 Redefine the restricted "at most one" quantifier context to avoid overloading and syntax check errors in mmj2. This operator does not require 𝑥 and 𝐵 disjoint.
wff ∃*𝑥 in 𝐵𝜑

Syntaxwl-reu 33980 Redefine the restricted existential uniqueness context to avoid overloading and syntax check errors in mmj2. This operator does not require 𝑥 and 𝐵 disjoint.
wff ∃!𝑥 in 𝐵𝜑

Syntaxwl-crab 33981 Redefine extended class notation to include the restricted class abstraction (class builder).
class {𝑥 in 𝐴𝜑}

Definitiondf-wl-ral 33982* Define an improved restricted universal quantifier. This version does not interpret elementhood verbatim in 𝑥 in 𝐴𝜑. Assuming a real elementhood leads to awkward consequences should the class 𝐴 depend on 𝑥. Instead a dummy variable 𝑦, disjoint from all other variables, is introduced to describe an element in 𝐴. The subexpression 𝑥(𝑥 = 𝑦𝜑) is [𝑦 / 𝑥]𝜑 in disguise (see wl-dfralsb 33983). This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurences in 𝜑 are fully honored.

If 𝑥 is not free in 𝐴, a simplification is possible ( see wl-dfralf 33985, wl-dfralv 33987). (Contributed by NM, 19-Aug-1993.) Isolate x from A, idea of Mario Carneiro. (Revised by Wolf Lammen, 21-May-2023.)

(∀𝑥 in 𝐴𝜑 ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-dfralsb 33983* An alternate definition of restricted universal quantification (df-wl-ral 33982) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
(∀𝑥 in 𝐴𝜑 ↔ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑))

Theoremwl-dfralflem 33984* Lemma for wl-dfralf 33985 and wl-dfralv . (Contributed by Wolf Lammen, 23-May-2023.)
(∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremwl-dfralf 33985 Restricted universal quantification (df-wl-ral 33982) allows a simplification, if we can assume all appearences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 23-May-2023.)
(𝑥𝐴 → (∀𝑥 in 𝐴𝜑 ↔ ∀𝑥(𝑥𝐴𝜑)))

Theoremwl-dfralfi 33986 Restricted universal quantification (df-wl-ral 33982) allows allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
𝑥𝐴       (∀𝑥 in 𝐴𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremwl-dfralv 33987* Alternate definition of restricted universal quantification (df-wl-ral ) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 23-May-2023.)
(∀𝑥 in 𝐴𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))

Definitiondf-wl-rex 33988 Define an improved restricted existential quantifier. This version does not interpret elementhood verbatim in 𝑥 in 𝐴𝜑. Assuming a real elementhood leads to awkward consequences should the class 𝐴 depend on 𝑥. Instead we base the definiton on df-wl-ral 33982, where this is already cleanly handled. Alternate definitions are wl-dfrexsb 33993 and wl-dfrexex 33992. If 𝑥 is not free in 𝐴, the defining expression can be simplified (see wl-dfrexf 33989, wl-dfrexv 33991).

This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurences in 𝜑 are fully honored. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 25-May-2023.)

(∃𝑥 in 𝐴𝜑 ↔ ¬ ∀𝑥 in 𝐴 ¬ 𝜑)

Theoremwl-dfrexf 33989 Restricted existential quantification (df-wl-rex 33988) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 25-May-2023.)
(𝑥𝐴 → (∃𝑥 in 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑)))

Theoremwl-dfrexfi 33990 Restricted universal quantification (df-wl-rex 33988) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
𝑥𝐴       (∃𝑥 in 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))

Theoremwl-dfrexv 33991* Alternate definition of restricted universal quantification (df-wl-rex 33988) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 25-May-2023.)
(∃𝑥 in 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))

Theoremwl-dfrexex 33992* Alternate definition of the restricted existential quantification (df-wl-rex 33988), according to the pattern given in df-wl-ral 33982. (Contributed by Wolf Lammen, 25-May-2023.)
(∃𝑥 in 𝐴𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-dfrexsb 33993* An alternate definition of restricted existential quantification (df-wl-rex 33988) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
(∃𝑥 in 𝐴𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))

Definitiondf-wl-rmo 33994* Define an improved restricted "at most one". This version does not interpret elementhood verbatim in ∃*𝑥 in 𝐴𝜑. Assuming a real elementhood leads to awkward consequences should the class 𝐴 depend on 𝑥. Instead we base the definiton on df-wl-ral 33982, where this is already cleanly handled.

This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurences in 𝜑 are fully honored.

Alternate definitions are given in wl-dfrmosb 33995 and, if 𝑥 is not free in 𝐴, wl-dfrmov 33996 and wl-dfrmof 33997. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 26-May-2023.)

(∃*𝑥 in 𝐴𝜑 ↔ ∃𝑦𝑥 in 𝐴(𝜑𝑥 = 𝑦))

Theoremwl-dfrmosb 33995* An alternate definition of restricted "at most one" (df-wl-rmo 33994) using substitution. (Contributed by Wolf Lammen, 27-May-2023.)
(∃*𝑥 in 𝐴𝜑 ↔ ∃*𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))

Theoremwl-dfrmov 33996* Alternate definition of restricted "at most one" (df-wl-rmo 33994) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
(∃*𝑥 in 𝐴𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))

Theoremwl-dfrmof 33997 Restricted "at most one" (df-wl-rmo 33994) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)
(𝑥𝐴 → (∃*𝑥 in 𝐴𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑)))

Definitiondf-wl-reu 33998 Define an improved restricted existential uniqueness. This version does not interpret elementhood verbatim in ∃!𝑥 in 𝐴𝜑. Assuming a real elementhood leads to awkward consequences should the class 𝐴 depend on 𝑥. Instead we base the definiton on df-wl-ral 33982, where this is already cleanly handled.

This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurences in 𝜑 are fully honored.

Alternate definitions are given in wl-dfreusb 33999 and, if 𝑥 is not free in 𝐴, wl-dfreuv 34000 and wl-dfreuf 34001. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 28-May-2023.)

(∃!𝑥 in 𝐴𝜑 ↔ (∃𝑥 in 𝐴𝜑 ∧ ∃*𝑥 in 𝐴𝜑))

Theoremwl-dfreusb 33999* An alternate definition of restricted existential uniqueness (df-wl-reu 33998) using substitution. (Contributed by Wolf Lammen, 28-May-2023.)
(∃!𝑥 in 𝐴𝜑 ↔ ∃!𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))

Theoremwl-dfreuv 34000* Alternate definition of restricted existential uniqueness (df-wl-reu 33998) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
(∃!𝑥 in 𝐴𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43671
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