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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ralab2 3001* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (x = y → (ψ ↔ χ)) ⇒ ⊢ (∀x ∈ {y ∣ φ}ψ ↔ ∀y(φ → χ)) | ||
Theorem | ralrab2 3002* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (x = y → (ψ ↔ χ)) ⇒ ⊢ (∀x ∈ {y ∈ A ∣ φ}ψ ↔ ∀y ∈ A (φ → χ)) | ||
Theorem | rexab2 3003* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (x = y → (ψ ↔ χ)) ⇒ ⊢ (∃x ∈ {y ∣ φ}ψ ↔ ∃y(φ ∧ χ)) | ||
Theorem | rexrab2 3004* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (x = y → (ψ ↔ χ)) ⇒ ⊢ (∃x ∈ {y ∈ A ∣ φ}ψ ↔ ∃y ∈ A (φ ∧ χ)) | ||
Theorem | abidnf 3005* | Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) |
⊢ (ℲxA → {z ∣ ∀x z ∈ A} = A) | ||
Theorem | dedhb 3006* | A deduction theorem for converting the inference ⊢ ℲxA => ⊢ φ into a closed theorem. Use nfa1 1788 and nfab 2493 to eliminate the hypothesis of the substitution instance ψ of the inference. For converting the inference form into a deduction form, abidnf 3005 is useful. (Contributed by NM, 8-Dec-2006.) |
⊢ (A = {z ∣ ∀x z ∈ A} → (φ ↔ ψ)) & ⊢ ψ ⇒ ⊢ (ℲxA → φ) | ||
Theorem | eqeu 3007* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ ((A ∈ B ∧ ψ ∧ ∀x(φ → x = A)) → ∃!xφ) | ||
Theorem | eueq 3008* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
⊢ (A ∈ V ↔ ∃!x x = A) | ||
Theorem | eueq1 3009* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ A ∈ V ⇒ ⊢ ∃!x x = A | ||
Theorem | eueq2 3010* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B)) | ||
Theorem | eueq3 3011* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
⊢ A ∈ V & ⊢ B ∈ V & ⊢ C ∈ V & ⊢ ¬ (φ ∧ ψ) ⇒ ⊢ ∃!x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)) | ||
Theorem | moeq 3012* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
⊢ ∃*x x = A | ||
Theorem | moeq3 3013* | "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.) |
⊢ B ∈ V & ⊢ C ∈ V & ⊢ ¬ (φ ∧ ψ) ⇒ ⊢ ∃*x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)) | ||
Theorem | mosub 3014* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*xφ ⇒ ⊢ ∃*x∃y(y = A ∧ φ) | ||
Theorem | mo2icl 3015* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
⊢ (∀x(φ → x = A) → ∃*xφ) | ||
Theorem | mob2 3016* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ ((A ∈ B ∧ ∃*xφ ∧ φ) → (x = A ↔ ψ)) | ||
Theorem | moi2 3017* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ (((A ∈ B ∧ ∃*xφ) ∧ (φ ∧ ψ)) → x = A) | ||
Theorem | mob 3018* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
⊢ (x = A → (φ ↔ ψ)) & ⊢ (x = B → (φ ↔ χ)) ⇒ ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ)) | ||
Theorem | moi 3019* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
⊢ (x = A → (φ ↔ ψ)) & ⊢ (x = B → (φ ↔ χ)) ⇒ ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ (ψ ∧ χ)) → A = B) | ||
Theorem | morex 3020* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ B ∈ V & ⊢ (x = B → (φ ↔ ψ)) ⇒ ⊢ ((∃x ∈ A φ ∧ ∃*xφ) → (ψ → B ∈ A)) | ||
Theorem | euxfr2 3021* | Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.) |
⊢ A ∈ V & ⊢ ∃*y x = A ⇒ ⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ) | ||
Theorem | euxfr 3022* | Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.) |
⊢ A ∈ V & ⊢ ∃!y x = A & ⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ (∃!xφ ↔ ∃!yψ) | ||
Theorem | euind 3023* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
⊢ B ∈ V & ⊢ (x = y → (φ ↔ ψ)) & ⊢ (x = y → A = B) ⇒ ⊢ ((∀x∀y((φ ∧ ψ) → A = B) ∧ ∃xφ) → ∃!z∀x(φ → z = A)) | ||
Theorem | reu2 3024* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y))) | ||
Theorem | reu6 3025* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
⊢ (∃!x ∈ A φ ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y)) | ||
Theorem | reu3 3026* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃y ∈ A ∀x ∈ A (φ → x = y))) | ||
Theorem | reu6i 3027* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ((B ∈ A ∧ ∀x ∈ A (φ ↔ x = B)) → ∃!x ∈ A φ) | ||
Theorem | eqreu 3028* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (x = B → (φ ↔ ψ)) ⇒ ⊢ ((B ∈ A ∧ ψ ∧ ∀x ∈ A (φ → x = B)) → ∃!x ∈ A φ) | ||
Theorem | rmo4 3029* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y)) | ||
Theorem | reu4 3030* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y))) | ||
Theorem | reu7 3031* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃x ∈ A ∀y ∈ A (ψ → x = y))) | ||
Theorem | reu8 3032* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!x ∈ A φ ↔ ∃x ∈ A (φ ∧ ∀y ∈ A (ψ → x = y))) | ||
Theorem | reueq 3033* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ (B ∈ A ↔ ∃!x ∈ A x = B) | ||
Theorem | rmoan 3034 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*x ∈ A φ → ∃*x ∈ A (ψ ∧ φ)) | ||
Theorem | rmoim 3035 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀x ∈ A (φ → ψ) → (∃*x ∈ A ψ → ∃*x ∈ A φ)) | ||
Theorem | rmoimia 3036 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (x ∈ A → (φ → ψ)) ⇒ ⊢ (∃*x ∈ A ψ → ∃*x ∈ A φ) | ||
Theorem | rmoimi2 3037 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ ψ)) ⇒ ⊢ (∃*x ∈ B ψ → ∃*x ∈ A φ) | ||
Theorem | 2reuswap 3038* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
⊢ (∀x ∈ A ∃*y ∈ B φ → (∃!x ∈ A ∃y ∈ B φ → ∃!y ∈ B ∃x ∈ A φ)) | ||
Theorem | reuind 3039* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
⊢ (x = y → (φ ↔ ψ)) & ⊢ (x = y → A = B) ⇒ ⊢ ((∀x∀y(((A ∈ C ∧ φ) ∧ (B ∈ C ∧ ψ)) → A = B) ∧ ∃x(A ∈ C ∧ φ)) → ∃!z ∈ C ∀x((A ∈ C ∧ φ) → z = A)) | ||
Theorem | 2rmorex 3040* | Double restricted quantification with "at most one," analogous to 2moex 2275. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y ∈ B ∃*x ∈ A φ) | ||
Theorem | 2reu5lem1 3041* | Lemma for 2reu5 3044. Note that ∃!x ∈ A∃!y ∈ Bφ does not mean "there is exactly one x in A and exactly one y in B such that φ holds;" see comment for 2eu5 2288. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃!x ∈ A ∃!y ∈ B φ ↔ ∃!x∃!y(x ∈ A ∧ y ∈ B ∧ φ)) | ||
Theorem | 2reu5lem2 3042* | Lemma for 2reu5 3044. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀x ∈ A ∃*y ∈ B φ ↔ ∀x∃*y(x ∈ A ∧ y ∈ B ∧ φ)) | ||
Theorem | 2reu5lem3 3043* | Lemma for 2reu5 3044. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3131. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ((∃!x ∈ A ∃!y ∈ B φ ∧ ∀x ∈ A ∃*y ∈ B φ) ↔ (∃x ∈ A ∃y ∈ B φ ∧ ∃z∃w∀x ∈ A ∀y ∈ B (φ → (x = z ∧ y = w)))) | ||
Theorem | 2reu5 3044* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2288 and reu3 3026. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ((∃!x ∈ A ∃!y ∈ B φ ∧ ∀x ∈ A ∃*y ∈ B φ) ↔ (∃x ∈ A ∃y ∈ B φ ∧ ∃z ∈ A ∃w ∈ B ∀x ∈ A ∀y ∈ B (φ → (x = z ∧ y = w)))) | ||
Theorem | ru 3045 |
Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as A ∈ V, asserted that any collection of sets A is a set i.e. belongs to the universe V of all sets. In particular, by substituting {x ∣ x ∉ x} (the "Russell class") for A, it asserted {x ∣ x ∉ x} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {x ∣ x ∉ x} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex in set.mm asserting that A is a set only when it is smaller than some other set B. However, Zermelo was then faced with a "chicken and egg" problem of how to show B is a set, leading him to introduce the set-building axioms of Null Set 0ex 4110, Pairing prex 4112, Union uniex 4317, Power Set pwex 4329, and Infinity omex in set.mm to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex in set.mm (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 in set.mm and Cantor's Theorem canth in set.mm are provably false! (See ncanth in set.mm for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep in set.mm replaces ax-rep in set.mm) with ax-sep restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944). Under ZF set theory, every set is a member of the Russell class by elirrv in set.mm (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv in set.mm). See ruALT in set.mm for an alternate proof of ru 3045 derived from that fact. (Contributed by NM, 7-Aug-1994.) |
⊢ {x ∣ x ∉ x} ∉ V | ||
Syntax | wsbc 3046 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class A for setvar variable x in wff φ." |
wff [̣A / x]̣φ | ||
Definition | df-sbc 3047 |
Define the proper substitution of a class for a set.
When A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3072 for our definition, which always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3048 below). For example, if A is a proper class, Quine's substitution of A for y in 0 ∈ y evaluates to 0 ∈ A rather than our falsehood. (This can be seen by substituting A, y, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactical breakdown of φ, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3048, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3047 in the form of sbc8g 3053. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of A in every use of this definition) we allow direct reference to df-sbc 3047 and assert that [̣A / x]̣φ is always false when A is a proper class. The theorem sbc2or 3054 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3048. The related definition df-csb 3137 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.) |
⊢ ([̣A / x]̣φ ↔ A ∈ {x ∣ φ}) | ||
Theorem | dfsbcq 3048 |
This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
under both our definition and Quine's, provides us with a weak definition
of the proper substitution of a class for a set. Since our df-sbc 3047 does
not result in the same behavior as Quine's for proper classes, if we
wished to avoid conflict with Quine's definition we could start with this
theorem and dfsbcq2 3049 instead of df-sbc 3047. (dfsbcq2 3049 is needed because
unlike Quine we do not overload the df-sb 1649 syntax.) As a consequence of
these theorems, we can derive sbc8g 3053, which is a weaker version of
df-sbc 3047 that leaves substitution undefined when A is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3053, so we will allow direct use of df-sbc 3047 after theorem sbc2or 3054 below. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) |
⊢ (A = B → ([̣A / x]̣φ ↔ [̣B / x]̣φ)) | ||
Theorem | dfsbcq2 3049 | This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1649 and substitution for class variables df-sbc 3047. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3048. (Contributed by NM, 31-Dec-2016.) |
⊢ (y = A → ([y / x]φ ↔ [̣A / x]̣φ)) | ||
Theorem | sbsbc 3050 | Show that df-sb 1649 and df-sbc 3047 are equivalent when the class term A in df-sbc 3047 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1649 for proofs involving df-sbc 3047. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([y / x]φ ↔ [̣y / x]̣φ) | ||
Theorem | sbceq1d 3051 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → ([̣A / x]̣φ ↔ [̣B / x]̣φ)) | ||
Theorem | sbceq1dd 3052 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (φ → A = B) & ⊢ (φ → [̣A / x]̣φ) ⇒ ⊢ (φ → [̣B / x]̣φ) | ||
Theorem | sbc8g 3053 | This is the closest we can get to df-sbc 3047 if we start from dfsbcq 3048 (see its comments) and dfsbcq2 3049. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
⊢ (A ∈ V → ([̣A / x]̣φ ↔ A ∈ {x ∣ φ})) | ||
Theorem | sbc2or 3054* | The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [A / x]φ behavior at proper classes, matching the sbc5 3070 (false) and sbc6 3072 (true) conclusions. This is interesting since dfsbcq 3048 and dfsbcq2 3049 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable y that φ or A may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.) |
⊢ (([̣A / x]̣φ ↔ ∃x(x = A ∧ φ)) ∨ ([̣A / x]̣φ ↔ ∀x(x = A → φ))) | ||
Theorem | sbcex 3055 | By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ ([̣A / x]̣φ → A ∈ V) | ||
Theorem | sbceq1a 3056 | Equality theorem for class substitution. Class version of sbequ12 1919. (Contributed by NM, 26-Sep-2003.) |
⊢ (x = A → (φ ↔ [̣A / x]̣φ)) | ||
Theorem | sbceq2a 3057 | Equality theorem for class substitution. Class version of sbequ12r 1920. (Contributed by NM, 4-Jan-2017.) |
⊢ (A = x → ([̣A / x]̣φ ↔ φ)) | ||
Theorem | spsbc 3058 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 2024 and rspsbc 3124. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ V → (∀xφ → [̣A / x]̣φ)) | ||
Theorem | spsbcd 3059 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 2024 and rspsbc 3124. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (φ → A ∈ V) & ⊢ (φ → ∀xψ) ⇒ ⊢ (φ → [̣A / x]̣ψ) | ||
Theorem | sbcth 3060 | A substitution into a theorem remains true (when A is a set). (Contributed by NM, 5-Nov-2005.) |
⊢ φ ⇒ ⊢ (A ∈ V → [̣A / x]̣φ) | ||
Theorem | sbcthdv 3061* | Deduction version of sbcth 3060. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (φ → ψ) ⇒ ⊢ ((φ ∧ A ∈ V) → [̣A / x]̣ψ) | ||
Theorem | sbcid 3062 | An identity theorem for substitution. See sbid 1922. (Contributed by Mario Carneiro, 18-Feb-2017.) |
⊢ ([̣x / x]̣φ ↔ φ) | ||
Theorem | nfsbc1d 3063 | Deduction version of nfsbc1 3064. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ (φ → ℲxA) ⇒ ⊢ (φ → Ⅎx[̣A / x]̣ψ) | ||
Theorem | nfsbc1 3064 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ ℲxA ⇒ ⊢ Ⅎx[̣A / x]̣φ | ||
Theorem | nfsbc1v 3065* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎx[̣A / x]̣φ | ||
Theorem | nfsbcd 3066 | Deduction version of nfsbc 3067. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎyφ & ⊢ (φ → ℲxA) & ⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx[̣A / y]̣ψ) | ||
Theorem | nfsbc 3067 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ ℲxA & ⊢ Ⅎxφ ⇒ ⊢ Ⅎx[̣A / y]̣φ | ||
Theorem | sbcco 3068* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ([̣A / y]̣[̣y / x]̣φ ↔ [̣A / x]̣φ) | ||
Theorem | sbcco2 3069* | A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (x = y → A = B) ⇒ ⊢ ([̣x / y]̣[̣B / x]̣φ ↔ [̣A / x]̣φ) | ||
Theorem | sbc5 3070* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ)) | ||
Theorem | sbc6g 3071* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∀x(x = A → φ))) | ||
Theorem | sbc6 3072* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
⊢ A ∈ V ⇒ ⊢ ([̣A / x]̣φ ↔ ∀x(x = A → φ)) | ||
Theorem | sbc7 3073* | An equivalence for class substitution in the spirit of df-clab 2340. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ([̣A / x]̣φ ↔ ∃y(y = A ∧ [̣y / x]̣φ)) | ||
Theorem | cbvsbc 3074 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ ([̣A / x]̣φ ↔ [̣A / y]̣ψ) | ||
Theorem | cbvsbcv 3075* | Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ ([̣A / x]̣φ ↔ [̣A / y]̣ψ) | ||
Theorem | sbciegft 3076* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3077.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → ([̣A / x]̣φ ↔ ψ)) | ||
Theorem | sbciegf 3077* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎxψ & ⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ)) | ||
Theorem | sbcieg 3078* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ)) | ||
Theorem | sbcie2g 3079* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3080 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (x = y → (φ ↔ ψ)) & ⊢ (y = A → (ψ ↔ χ)) ⇒ ⊢ (A ∈ V → ([̣A / x]̣φ ↔ χ)) | ||
Theorem | sbcie 3080* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
⊢ A ∈ V & ⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ ([̣A / x]̣φ ↔ ψ) | ||
Theorem | sbciedf 3081* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
⊢ (φ → A ∈ V) & ⊢ ((φ ∧ x = A) → (ψ ↔ χ)) & ⊢ Ⅎxφ & ⊢ (φ → Ⅎxχ) ⇒ ⊢ (φ → ([̣A / x]̣ψ ↔ χ)) | ||
Theorem | sbcied 3082* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
⊢ (φ → A ∈ V) & ⊢ ((φ ∧ x = A) → (ψ ↔ χ)) ⇒ ⊢ (φ → ([̣A / x]̣ψ ↔ χ)) | ||
Theorem | sbcied2 3083* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
⊢ (φ → A ∈ V) & ⊢ (φ → A = B) & ⊢ ((φ ∧ x = B) → (ψ ↔ χ)) ⇒ ⊢ (φ → ([̣A / x]̣ψ ↔ χ)) | ||
Theorem | elrabsf 3084 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2993 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
⊢ ℲxB ⇒ ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ [̣A / x]̣φ)) | ||
Theorem | eqsbc3 3085* | Substitution applied to an atomic wff. Set theory version of eqsb3 2454. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ (A ∈ V → ([̣A / x]̣x = B ↔ A = B)) | ||
Theorem | sbcng 3086 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ V → ([̣A / x]̣ ¬ φ ↔ ¬ [̣A / x]̣φ)) | ||
Theorem | sbcimg 3087 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ V → ([̣A / x]̣(φ → ψ) ↔ ([̣A / x]̣φ → [̣A / x]̣ψ))) | ||
Theorem | sbcan 3088 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
⊢ ([̣A / x]̣(φ ∧ ψ) ↔ ([̣A / x]̣φ ∧ [̣A / x]̣ψ)) | ||
Theorem | sbcang 3089 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
⊢ (A ∈ V → ([̣A / x]̣(φ ∧ ψ) ↔ ([̣A / x]̣φ ∧ [̣A / x]̣ψ))) | ||
Theorem | sbcor 3090 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
⊢ ([̣A / x]̣(φ ∨ ψ) ↔ ([̣A / x]̣φ ∨ [̣A / x]̣ψ)) | ||
Theorem | sbcorg 3091 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
⊢ (A ∈ V → ([̣A / x]̣(φ ∨ ψ) ↔ ([̣A / x]̣φ ∨ [̣A / x]̣ψ))) | ||
Theorem | sbcbig 3092 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ (A ∈ V → ([̣A / x]̣(φ ↔ ψ) ↔ ([̣A / x]̣φ ↔ [̣A / x]̣ψ))) | ||
Theorem | sbcal 3093* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
⊢ ([̣A / y]̣∀xφ ↔ ∀x[̣A / y]̣φ) | ||
Theorem | sbcalg 3094* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ V → ([̣A / y]̣∀xφ ↔ ∀x[̣A / y]̣φ)) | ||
Theorem | sbcex2 3095* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
⊢ ([̣A / y]̣∃xφ ↔ ∃x[̣A / y]̣φ) | ||
Theorem | sbcexg 3096* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
⊢ (A ∈ V → ([̣A / y]̣∃xφ ↔ ∃x[̣A / y]̣φ)) | ||
Theorem | sbceqal 3097* | Set theory version of sbeqal1 in set.mm. (Contributed by Andrew Salmon, 28-Jun-2011.) |
⊢ (A ∈ V → (∀x(x = A → x = B) → A = B)) | ||
Theorem | sbeqalb 3098* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
⊢ (A ∈ V → ((∀x(φ ↔ x = A) ∧ ∀x(φ ↔ x = B)) → A = B)) | ||
Theorem | sbcbid 3099 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ([̣A / x]̣ψ ↔ [̣A / x]̣χ)) | ||
Theorem | sbcbidv 3100* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ([̣A / x]̣ψ ↔ [̣A / x]̣χ)) |
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