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Type | Label | Description |
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Statement | ||
Theorem | 2rexreu 3701* | Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2708. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
This is a very useless definition, which "abbreviates" (𝑥 = 𝑦 → 𝜑) as CondEq(𝑥 = 𝑦 → 𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation (𝑥 = 𝑦 → 𝜑). This is all used as part of a metatheorem: we want to say that ⊢ (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and ⊢ (𝑥 = 𝑦 → 𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables assumption: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦 → 𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜑)), which is provable by cdeqth 3706 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one setvar variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the universal quantifier and the class builder). In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 3711 and cdeqab 3709. | ||
Syntax | wcdeq 3702 | Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result. |
wff CondEq(𝑥 = 𝑦 → 𝜑) | ||
Definition | df-cdeq 3703 | Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | ||
Theorem | cdeqi 3704 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqri 3705 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqth 3706 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝜑 ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqnot 3707 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) | ||
Theorem | cdeqal 3708* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
Theorem | cdeqab 3709* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) | ||
Theorem | cdeqal1 3710* | Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | cdeqab1 3711* | Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) | ||
Theorem | cdeqim 3712 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | cdeqcv 3713 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | cdeqeq 3714 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | cdeqel 3715 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | nfcdeq 3716* | If we have a conditional equality proof, where 𝜑 is 𝜑(𝑥) and 𝜓 is 𝜑(𝑦), and 𝜑(𝑥) in fact does not have 𝑥 free in it according to Ⅎ, then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that Ⅎ𝑥𝜑 is actually a not-free predicate. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜑 ↔ 𝜓) | ||
Theorem | nfccdeq 3717* | Variation of nfcdeq 3716 for classes. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2158. (Revised by Gino Giotto, 19-May-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | rru 3718* |
Relative version of Russell's paradox ru 3719 (which corresponds to the
case 𝐴 = V).
Originally a subproof in pwnss 5215. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3092. (Revised by Steven Nguyen, 23-Nov-2022.) |
⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 | ||
Theorem | ru 3719 |
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, which Frege acknowledged in the second edition of his Grundgesetze der Arithmetik. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 5189 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 5175, Pairing prex 5298, Union uniex 7447, Power Set pwex 5246, and Infinity omex 9090 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 6411 (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, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ac4 9886 and Cantor's theorem canth 7090 are provably false. (See ncanth 7091 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 5167 replaces ax-rep 5154) with ax-sep 5167 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 our ZF set theory, every set is a member of the Russell class by elirrv 9044 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9050). See ruALT 9051 for an alternate proof of ru 3719 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2379. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Syntax | wsbc 3720 | 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 3721 |
Define the proper substitution of a class for a set.
When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3730). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3750 for our definition, whose right-hand side 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 3722 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 syntactic 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 3722, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3721 in the form of sbc8g 3728. 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 3721 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3729 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3722. The related definition df-csb 3829 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 3722 |
Proper substitution of a class for a set in a wff given equal classes.
This is the essence of the sixth axiom of Frege, specifically Proposition
52 of [Frege1879] p. 50.
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 3721 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 3723 instead of df-sbc 3721. (dfsbcq2 3723 is needed because unlike Quine we do not overload the df-sb 2070 syntax.) As a consequence of these theorems, we can derive sbc8g 3728, which is a weaker version of df-sbc 3721 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 3728, so we will allow direct use of df-sbc 3721 after theorem sbc2or 3729 below. Proper substitution 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 3723 | 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 2070 and substitution for class variables df-sbc 3721. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3722. (Contributed by NM, 31-Dec-2016.) |
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbsbc 3724 | Show that df-sb 2070 and df-sbc 3721 are equivalent when the class term 𝐴 in df-sbc 3721 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2070 for proofs involving df-sbc 3721. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbceq1d 3725 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | ||
Theorem | sbceq1dd 3726 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | ||
Theorem | sbceqbid 3727* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | ||
Theorem | sbc8g 3728 | This is the closest we can get to df-sbc 3721 if we start from dfsbcq 3722 (see its comments) and dfsbcq2 3723. (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 3729* | 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 3748 (false) and sbc6 3750 (true) conclusions. This is interesting since dfsbcq 3722 and dfsbcq2 3723 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not 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 3730 | 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 3731 | Equality theorem for class substitution. Class version of sbequ12 2250. (Contributed by NM, 26-Sep-2003.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbceq2a 3732 | Equality theorem for class substitution. Class version of sbequ12r 2251. (Contributed by NM, 4-Jan-2017.) |
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | spsbc 3733 | 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. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2073 and rspsbc 3808. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | ||
Theorem | spsbcd 3734 | 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 2073 and rspsbc 3808. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcth 3735 | 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 3736* | Deduction version of sbcth 3735. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcid 3737 | An identity theorem for substitution. See sbid 2254. (Contributed by Mario Carneiro, 18-Feb-2017.) |
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | nfsbc1d 3738 | Deduction version of nfsbc1 3739. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) | ||
Theorem | nfsbc1 3739 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbc1v 3740* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbcdw 3741* | Deduction version of nfsbcw 3742. Version of nfsbcd 3744 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | ||
Theorem | nfsbcw 3742* | Bound-variable hypothesis builder for class substitution. Version of nfsbc 3745 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 7-Sep-2014.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 | ||
Theorem | sbccow 3743* | A composition law for class substitution. Version of sbcco 3746 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | nfsbcd 3744 | Deduction version of nfsbc 3745. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfsbcdw 3741 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | ||
Theorem | nfsbc 3745 | Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfsbcw 3742 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 | ||
Theorem | sbcco 3746* | A composition law for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker sbccow 3743 when possible. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.) |
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcco2 3747* | 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 3748* | 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 3749* | 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 3750* | 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 3751* | An equivalence for class substitution in the spirit of df-clab 2777. 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 | cbvsbcw 3752* | Change bound variables in a wff substitution. Version of cbvsbc 3754 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Jeff Hankins, 19-Sep-2009.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | cbvsbcvw 3753* | Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3755 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 30-Sep-2008.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | cbvsbc 3754 | Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvsbcw 3752 when possible. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | cbvsbcv 3755* | Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvsbcvw 3753 when possible. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | sbciegft 3756* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3757.) (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 3757* | 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 3758* | 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 3759* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3760 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 3760* | 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 3761* | 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 3762* | 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 3763* | 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 3764 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3624 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 3765* | Substitution applied to an atomic wff. Class version of eqsb3 2916. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2379. (Revised by Wolf Lammen, 29-Apr-2023.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | sbcng 3766 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbcimg 3767 | 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 3768 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcor 3769 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbig 3770 | 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 | sbcn1 3771 | Move negation in and out of class substitution. One direction of sbcng 3766 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcim1 3772 | Distribution of class substitution over implication. One direction of sbcimg 3767 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbid 3773 | Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcbidv 3774* | Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) Drop ax-12 2175. (Revised by Gino Giotto, 1-Dec-2023.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcbidvOLD 3775* | Obsolete version of sbcbidv 3774 as of 1-Dec-2023. (Contributed by NM, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcbii 3776 | Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcbi1 3777 | Distribution of class substitution over biconditional. One direction of sbcbig 3770 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbi2 3778 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) Avoid ax-10, ax-12. (Revised by Steven Nguyen, 5-May-2024.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbi2OLD 3779 | Obsolete proof of sbcbi2 3778 as of 5-May-2024. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcal 3780* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑) | ||
Theorem | sbcex2 3781* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) | ||
Theorem | sbceqal 3782* | Class version of one implication of equvelv 2038. (Contributed by Andrew Salmon, 28-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | ||
Theorem | sbeqalb 3783* | 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 | eqsbc3r 3784* | eqsbc3 3765 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) | ||
Theorem | sbc3an 3785 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓 ∧ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcel1v 3786* | Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2379. (Revised by Wolf Lammen, 30-Apr-2023.) |
⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | sbcel2gv 3787* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | sbcel21v 3788* | Class substitution into a membership relation. One direction of sbcel2gv 3787 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵) | ||
Theorem | sbcimdv 3789* | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1812). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbctt 3790 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbcgf 3791 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbc19.21g 3792 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) | ||
Theorem | sbcg 3793* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3791. (Contributed by Alan Sare, 10-Nov-2012.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbcgfi 3794 | Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbc2iegf 3795* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊 & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) | ||
Theorem | sbc2ie 3796* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) | ||
Theorem | sbc2iedv 3797* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) | ||
Theorem | sbc3ie 3798* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) | ||
Theorem | sbccomlem 3799* | Lemma for sbccom 3800. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |
⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
Theorem | sbccom 3800* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑) |
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