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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-sbex 32601 If a proposition is true for a specific instance, then there exists an instance such that it is true for it. Uses only ax-1--6. Compare spsbe 1882 which, due to the specific form of df-sb 1879, uses fewer axioms. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)
 
Theorembj-ssbeq 32602* Substitution in an equality, disjoint variables case. Uses only ax-1--6. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 32602 first with a DV on x,t, and then in the general case. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝑦 = 𝑧𝑦 = 𝑧)
 
Theorembj-ssb0 32603* Substitution for a variable not occurring in a proposition. See sbf 2378. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑𝜑)
 
Theorembj-ssbequ 32604 Equality property for substitution, from Tarski's system. Compare sbequ 2374. (Contributed by BJ, 30-Dec-2020.)
(𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑))
 
Theorembj-ssblem1 32605* A lemma for the definiens of df-sb 1879. An instance of sp 2051 proved without it. Note: it has a common subproof with bj-ssbjust 32593. (Contributed by BJ, 22-Dec-2020.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-ssblem2 32606* An instance of ax-11 2032 proved without it. The converse may not be provable without ax-11 2032 (since using alcomiw 1969 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
 
Theorembj-ssb1a 32607* One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 32608 for the biconditional, which requires ax-11 2032. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)
 
Theorembj-ssb1 32608* A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 32607 for the backward implication, which does not require ax-11 2032 (note that here, the version of ax-11 2032 with disjoint setvar metavariables would suffice). Compare sb6 2427. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
 
Theorembj-ax12 32609* A weaker form of ax-12 2045 and ax12v2 2047, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.)
𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
 
Theorembj-ax12ssb 32610* The axiom bj-ax12 32609 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
[𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)
 
Theorembj-modal5e 32611 Dual statement of hbe1 2019 (which is the real modal-5 2030). See also axc7 2130 and axc7e 2131. (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)
 
Theorembj-19.41al 32612 Special case of 19.41 2101 proved from Tarski, ax-10 2017 (modal5) and hba1 2149 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
 
Theorembj-equsexval 32613* Special case of equsexv 2107 proved from Tarski, ax-10 2017 (modal5) and hba1 2149 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)
 
Theorembj-sb56 32614* Proof of sb56 2148 from Tarski, ax-10 2017 (modal5) and bj-ax12 32609. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
Theorembj-dfssb2 32615* An alternate definition of df-ssb 32595. Note that the use of a dummy variable in the definition df-ssb 32595 allows to use bj-sb56 32614 instead of equs45f 2348 and hence to avoid dependency on ax-13 2244 and to use ax-12 2045 only through bj-ax12 32609. Compare dfsb7 2453. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-ssbn 32616 The result of a substitution in the negation of a formula is the negation of the result of the same substitution in that formula. Proved from Tarski, ax-10 2017, bj-ax12 32609. Compare sbn 2389. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑)
 
Theorembj-ssbft 32617 See sbft 2377. This proof is from Tarski's FOL together with sp 2051 (and its dual). (Contributed by BJ, 22-Dec-2020.)
(Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))
 
Theorembj-ssbequ2 32618 Note that ax-12 2045 is used only via sp 2051. See sbequ2 1880 and stdpc7 1956. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))
 
Theorembj-ssbequ1 32619 This uses ax-12 2045 with a direct reference to ax12v 2046. Therefore, compared to bj-ax12 32609, there is a hidden use of sp 2051. Note that with ax-12 2045, it can be proved with dv condition on 𝑥, 𝑡. See sbequ1 2108. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))
 
Theorembj-ssbid2 32620 A special case of bj-ssbequ2 32618. (Contributed by BJ, 22-Dec-2020.)
([𝑥/𝑥]b𝜑𝜑)
 
Theorembj-ssbid2ALT 32621 Alternate proof of bj-ssbid2 32620, not using bj-ssbequ2 32618. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥/𝑥]b𝜑𝜑)
 
Theorembj-ssbid1 32622 A special case of bj-ssbequ1 32619. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥/𝑥]b𝜑)
 
Theorembj-ssbid1ALT 32623 Alternate proof of bj-ssbid1 32622, not using bj-ssbequ1 32619. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥/𝑥]b𝜑)
 
Theorembj-ssbssblem 32624* Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)
 
Theorembj-ssbcom3lem 32625* Lemma for bj-ssbcom3 when setvar variables are disjoint. Remark: does not seem useful. (Contributed by BJ, 30-Dec-2020.)
([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b[𝑡/𝑥]b𝜑)
 
Theorembj-ax6elem1 32626* Lemma for bj-ax6e 32628. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theorembj-ax6elem2 32627* Lemma for bj-ax6e 32628. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
 
Theorembj-ax6e 32628 Proof of ax6e 2248 (hence ax6 2249) from Tarski's system, ax-c9 33994, ax-c16 33996. Remark: ax-6 1886 is used only via its principal (unbundled) instance ax6v 1887. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦
 
20.14.4.5  Adding ax-6
 
Theorembj-extru 32629 There exists a variable such that holds; that is, there exists a variable. This corresponds under the standard translation to one of the formulations of the modal axiom (D), the other being 19.2 1890. (This is also extt 32378; propose to move to Main extt 32378 and allt 32375; relabel exiftru 1889 to "exgen", for "existential generalization", which is the standard name for that rule of inference ? ). (Contributed by BJ, 12-May-2019.) (Proof modification is discouraged.)
𝑥
 
Theorembj-alequexv 32630* Version of bj-alequex 32683 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
 
Theorembj-spimvwt 32631* Closed form of spimvw 1925. See also spimt 2251. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))
 
Theorembj-spimevw 32632* Existential introduction, using implicit substitution. This is to spimeh 1923 what spimvw 1925 is to spimw 1924. (Contributed by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theorembj-spnfw 32633 Theorem close to a closed form of spnfw 1926. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorembj-cbvexiw 32634* Change bound variable. This is to cbvexvw 1968 what cbvaliw 1931 is to cbvalvw 1967. [TODO: move after cbvalivw 1932]. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-cbvexivw 32635* Change bound variable. This is to cbvexvw 1968 what cbvalivw 1932 is to cbvalvw 1967. [TODO: move after cbvalivw 1932]. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-modald 32636 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
 
Theorembj-denot 32637* A weakening of ax-6 1886 and ax6v 1887. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
 
Theorembj-eqs 32638* A lemma for substitutions, proved from Tarski's FOL. The version without DV(𝑥, 𝑦) is true but requires ax-13 2244. The DV condition DV( 𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
20.14.4.6  Adding ax-7
 
Theorembj-cbvexw 32639* Change bound variable. This is to cbvexvw 1968 what cbvalw 1966 is to cbvalvw 1967. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
Theorembj-ax12w 32640* The general statement that ax12w 2008 proves. (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))       (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))
 
20.14.4.7  Membership predicate, ax-8 and ax-9
 
Theorembj-elequ2g 32641* A form of elequ2 2002 with a universal quantifier. Its converse is ax-ext 2600. (TODO: move to main part, minimize axext4 2604--- as of 4-Nov-2020, minimizes only axext4 2604, by 13 bytes; and link to it in the comment of ax-ext 2600.) (Contributed by BJ, 3-Oct-2019.)
(𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
 
Theorembj-ax89 32642 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1990 and ax-9 1997. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1990 and ax-9 1997, as proved here. In the other direction, one can prove ax-8 1990 (respectively ax-9 1997) from bj-ax89 32642 by using mpan2 706 ( respectively mpan 705) and equid 1937. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 
Theorembj-elequ12 32643 An identity law for the non-logical predicate, which combines elequ1 1995 and elequ2 2002. For the analogous theorems for class terms, see eleq1 2687, eleq2 2688 and eleq12 2689. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 
Theorembj-cleljusti 32644* One direction of cleljust 1996, requiring only ax-1 6-- ax-5 1837 and ax8v1 1992. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)
 
20.14.4.8  Adding ax-11
 
Theorembj-alcomexcom 32645 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1735 section, soon after 2nexaln 1755, and used to prove excom 2040. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))
 
Theorembj-hbalt 32646 Closed form of hbal 2034. When in main part, prove hbal 2034 and hbald 2039 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
 
20.14.4.9  Adding ax-12
 
Theoremaxc11n11 32647 Proof of axc11n 2305 from { ax-1 6-- ax-7 1933, axc11 2312 } . Almost identical to axc11nfromc11 34030. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaxc11n11r 32648 Proof of axc11n 2305 from { ax-1 6-- ax-7 1933, axc9 2300, axc11r 2185 } (note that axc16 2133 is provable from { ax-1 6-- ax-7 1933, axc11r 2185 }).

Note that axc11n 2305 proves (over minimal calculus) that axc11 2312 and axc11r 2185 are equivalent. Therefore, axc11n11 32647 and axc11n11r 32648 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2312 appears slightly stronger since axc11n11r 32648 requires axc9 2300 while axc11n11 32647 does not).

(Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theorembj-axc16g16 32649* Proof of axc16g 2132 from { ax-1 6-- ax-7 1933, axc16 2133 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
Theorembj-ax12v3 32650* A weak version of ax-12 2045 which is stronger than ax12v 2046. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 1937), then bj-ax12v3 32650 implies ax-5 1837 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 32651. (Contributed by BJ, 6-Jul-2021.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-ax12v3ALT 32651* Alternate proof of bj-ax12v3 32650. Uses axc11r 2185 and axc15 2301 instead of ax-12 2045. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-sb 32652* A weak variant of sbid2 2411 not requiring ax-13 2244 nor ax-10 2017. On top of Tarski's FOL, one implication requires only ax12v 2046, and the other requires only sp 2051. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-modalbe 32653 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2140. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)
 
Theorembj-spst 32654 Closed form of sps 2053. Once in main part, prove sps 2053 and spsd 2055 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorembj-19.21bit 32655 Closed form of 19.21bi 2057. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
 
Theorembj-19.23bit 32656 Closed form of 19.23bi 2059. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))
 
Theorembj-nexrt 32657 Closed form of nexr 2060. Contrapositive of 19.8a 2050. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)
 
Theorembj-alrim 32658 Closed form of alrimi 2080. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-alrim2 32659 Uncurried (imported) form of bj-alrim 32658. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))
 
Theorembj-nfdt0 32660 A theorem close to a closed form of nf5d 2116 and nf5dh 2024. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
 
Theorembj-nfdt 32661 Closed form of nf5d 2116 and nf5dh 2024. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
 
Theorembj-nexdt 32662 Closed form of nexd 2087. (Contributed by BJ, 20-Oct-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
 
Theorembj-nexdvt 32663* Closed form of nexdv 1862. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
 
Theorembj-19.3t 32664 Closed form of 19.3 2067. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑𝜑))
 
Theorembj-alexbiex 32665 Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-exexbiex 32666 Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-alalbial 32667 Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-exalbial 32668 Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-19.9htbi 32669 Strengthening 19.9ht 2141 by replacing its succedent with a biconditional (19.9t 2069 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
 
Theorembj-hbntbi 32670 Strengthening hbnt 2142 by replacing its succedent with a biconditional. See also hbntg 31685 and hbntal 38589. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 32669. (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
 
Theorembj-biexal1 32671 A general FOL biconditional that generalizes 19.9ht 2141 among others. For this and the following theorems, see also 19.35 1803, 19.21 2073, 19.23 2078. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
 
Theorembj-biexal2 32672 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
 
Theorembj-biexal3 32673 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))
 
Theorembj-bialal 32674 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theorembj-biexex 32675 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))
 
Theorembj-hbext 32676 Closed form of hbex 2154. (Contributed by BJ, 10-Oct-2019.)
(∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theorembj-nfalt 32677 Closed form of nfal 2151. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theorembj-nfext 32678 Closed form of nfex 2152. (Contributed by BJ, 10-Oct-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theorembj-eeanvw 32679* Version of eeanv 2180 with a DV condition on 𝑥, 𝑦 not requiring ax-11 2032. (The same can be done with eeeanv 2181 and ee4anv 2182.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theorembj-modal4e 32680 Dual statement of hba1 2149 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∃𝑥𝜑)
 
Theorembj-modalb 32681 A short form of the axiom B of modal logic. (Contributed by BJ, 4-Apr-2021.)
𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
20.14.4.10  Adding ax-13
 
Theorembj-axc10 32682 Alternate (shorter) proof of axc10 2250. One can prove a version with DV(x,y) without ax-13 2244, by using ax6ev 1888 instead of ax6e 2248. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 
Theorembj-alequex 32683 A fol lemma. See bj-alequexv 32630 for a version with a DV condition requiring fewer axioms. Can be used to reduce the proof of spimt 2251 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
 
Theorembj-spimt2 32684 A step in the proof of spimt 2251. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∃𝑥𝜓𝜓) → (∀𝑥𝜑𝜓)))
 
Theorembj-cbv3ta 32685 Closed form of cbv3 2263. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
 
Theorembj-cbv3tb 32686 Closed form of cbv3 2263. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
 
Theorembj-hbsb3t 32687 A theorem close to a closed form of hbsb3 2362. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
 
Theorembj-hbsb3 32688 Shorter proof of hbsb3 2362. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 
Theorembj-nfs1t 32689 A theorem close to a closed form of nfs1 2363. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
 
Theorembj-nfs1t2 32690 A theorem close to a closed form of nfs1 2363. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
 
Theorembj-nfs1 32691 Shorter proof of nfs1 2363 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
𝑦𝜑       𝑥[𝑦 / 𝑥]𝜑
 
20.14.4.11  Removing dependencies on ax-13 (and ax-11)

It is known that ax-13 2244 is logically redundant (see ax13w 2011 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2244 from every theorem in set.mm which is totally unbundled (i.e., has dv conditions on all setvar variables). Indeed, start with the existing proof, and replace any occurrence of ax-13 2244 with ax13w 2011.

This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2244 (and using ax6v 1887 / ax6ev 1888 instead of ax-6 1886 / ax6e 2248, as is currently done).

One reason to be optimistic is that the first few utility theorems using ax-13 2244 (roughly 200 of them) are then used mainly with dummy variables, which one can assume distinct from any other, so that the unbundled versions of the utility theorems suffice.

In this section, we prove versions of theorems in the main part with dv conditions and not requiring ax-13 2244, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1887 and ax6ev 1888 instead of ax-6 1886 and ax6e 2248, and ax-5 1837 instead of ax13v 2245; shorter proofs may be possible). When no additional dv condition is required, we label it bj-xxx.

It is important to keep all the bundled theorems already in set.mm, but one may also add the (partially) unbundled versions which dipense with ax-13 2244, so as to remove dependencies on ax-13 2244 from many existing theorems.

UPDATE: it turns out that several theorems of the form bj-xxxv, or minor variations, are already in set.mm with label xxxw.

It is also possible to remove dependencies on ax-11 2032, typically by replacing a non-free hypothesis with a dv condition (see bj-cbv3v2 32702 and following theorems).

 
Theorembj-axc10v 32692* Version of axc10 2250 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 
Theorembj-spimtv 32693* Version of spimt 2251 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
 
Theorembj-spimedv 32694* Version of spimed 2253 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))
 
Theorembj-spimev 32695* Version of spime 2254 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theorembj-spimvv 32696* Version of spimv 2255 and spimv1 2113 with a dv condition, which does not require ax-13 2244. UPDATE: this is spimvw 1925. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theorembj-spimevv 32697* Version of spimev 2257 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theorembj-spvv 32698* Version of spv 2258 with a dv condition, which does not require ax-7 1933, ax-12 2045, ax-13 2244. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theorembj-speiv 32699* Version of spei 2259 with a dv condition, which does not require ax-13 2244 (neither ax-7 1933 nor ax-12 2045). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑
 
Theorembj-chvarv 32700* Version of chvar 2260 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓
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