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Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-exalimi 31801 An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1865 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Theorembj-ax12ig 31802 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 31803. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12i 31803 A weakening of bj-ax12ig 31802 that is sufficient to prove a weak form of the axiom of substitution ax-12 2034. The general statement of which ax12i 1866 is an instance. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12iOLD 31804 Old proof of bj-ax12i 31803. Obsolete as of 29-Dec-2020. (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax5ea 31805* If a formula holds for some value of a variable not occurring in it, then it holds for all values of that variable. (Contributed by BJ, 28-Dec-2020.)
(∃𝑥𝜑 → ∀𝑥𝜑)

Theorembj-nfv 31806* A non-occurring variable is semantically non-free. (Contributed by BJ, 6-May-2019.)
ℲℲ𝑥𝜑

Theorembj-ax12wlem 31807* A lemma used to prove a weak version of the axiom of substitution ax-12 2034. (Temporary comment: The general statement that ax12wlem 1996 proves.) (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

21.14.4.5  Equality and substitution

Syntaxwssb 31808 Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.)
wff [𝑡/𝑥]b𝜑

Definitiondf-ssb 31809* Alternate definition of proper substitution. Note that the occurrences of a given variable are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols. It is obtained by applying twice Tarski's definition sb6 2417 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2443 with respect to sb5 2418.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a DV condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row.

A drawback compared with df-sb 1868 is that this definition uses a dummy variable and therefore requires a justification theorem, which requires some of the auxiliary axiom schemes.

Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2041, of the substitution axiom ax-12 2034, and of commutation of quantifiers ax-11 2021; that is, ax-13 2234 will often be avoided. (Contributed by BJ, 22-Dec-2020.)

([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbim 31810 Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2382. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Theorembj-ssbbi 31811 Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2390. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Theorembj-ssbimi 31812 Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)

Theorembj-ssbbii 31813 Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)

Theorembj-alsb 31814 If a proposition is true for all instances, then it is true for any specific one. See stdpc4 2341. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-sbex 31815 If a proposition is true for a specific instance, then there exists an instance such that it is true for it. See spsbe 1871. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)

Theorembj-ssbeq 31816* Substitution in an equality, disjoint variables case. Might be shorter to prove the result about composition of two substitutions and prove this 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 31817* Substitution for a variable not occurring in a proposition. See sbf 2368. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑𝜑)

Theorembj-ssbequ 31818 Equality property for substitution, from Tarski's system. Compare sbequ 2364. (Contributed by BJ, 30-Dec-2020.)
(𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑))

Theorembj-ssblem1 31819* A lemma for the definiens of df-sb 1868. (Contributed by BJ, 22-Dec-2020.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssblem2 31820* The converse may not be provable without ax-11 2021. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))

Theorembj-ssb1a 31821* One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 31822 for the biconditional, which requires ax-11 2021. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)

Theorembj-ssb1 31822* A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 31821 for the backward implication, which does not require ax-11 2021 (note that here, the version of ax-11 2021 with disjoint setvar metavariables would suffice). Compare sb6 2417. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))

Theorembj-ax12 31823* A weaker form of ax-12 2034 and ax12v2 2036, 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 31824* The axiom bj-ax12 31823 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
[𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-modal5e 31825 Dual statement of hbe1 2008 (which is the real modal-5 2019). See also axc7 2117 and axc7e 2118. (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Theorembj-19.41al 31826 Special case of 19.41 2090 proved from Tarski, ax-10 2006 (modal5) and hba1 2137 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Theorembj-equsexval 31827* Special case of equsexv 2095 proved from Tarski, ax-10 2006 (modal5) and hba1 2137 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)

Theorembj-sb56 31828* Proof of sb56 2136 from Tarski, ax-10 2006 (modal5) and bj-ax12 31823. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-dfssb2 31829* An alternate definition of df-ssb 31809. Note that the use of a dummy variable in the definition df-ssb 31809 allows to use bj-sb56 31828 instead of equs45f 2338 and hence to avoid dependency on ax-13 2234 and to use ax-12 2034 only through bj-ax12 31823. Compare dfsb7 2443. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbn 31830 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 2006, bj-ax12 31823. Compare sbn 2379. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑)

Theorembj-ssbft 31831 See sbft 2367. This proof is from Tarski's FOL together with sp 2041 (and its dual). (Contributed by BJ, 22-Dec-2020.)
(ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ2 31832 Note that ax-12 2034 is used only via sp 2041. See sbequ2 1869 and stdpc7 1945. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ1 31833 This uses ax-12 2034 with a direct reference to ax12v 2035. Therefore, compared to bj-ax12 31823, there is a hidden use of sp 2041. Note that with ax-12 2034, it can be proved with dv condition on 𝑥, 𝑡. See sbequ1 2096. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))

Theorembj-ssbid2 31834 A special case of bj-ssbequ2 31832. (Contributed by BJ, 22-Dec-2020.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid2ALT 31835 Alternate proof of bj-ssbid2 31834, not using bj-ssbequ2 31832. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid1 31836 A special case of bj-ssbequ1 31833. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbid1ALT 31837 Alternate proof of bj-ssbid1 31836, not using bj-ssbequ1 31833. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbssblem 31838* 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 31839* 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 31840* Lemma for bj-ax6e 31842. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theorembj-ax6elem2 31841* Lemma for bj-ax6e 31842. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Theorembj-ax6e 31842 Proof of ax6e 2238 (hence ax6 2239) from Tarski's system, ax-c9 33193, ax-c16 33195. Remark: ax-6 1875 is used only via its principal (unbundled) instance ax6v 1876. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theorembj-extru 31843 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 1879. (This is also extt 31573; propose to move to Main extt 31573 and allt 31570; relabel exiftru 1878 to exgen ? ). (Contributed by BJ, 12-May-2019.) (Proof modification is discouraged.)
𝑥

Theorembj-spimevw 31844* Existential introduction, using implicit substitution. This is to spimeh 1912 what spimvw 1914 is to spimw 1913. (Contributed by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spnfw 31845 Theorem close to a closed form of spnfw 1915. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-cbvexiw 31846* Change bound variable. This is to cbvexvw 1957 what cbvaliw 1920 is to cbvalvw 1956. [TODO: move after cbvalivw 1921]. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbvexivw 31847* Change bound variable. This is to cbvexvw 1957 what cbvalivw 1921 is to cbvalvw 1956. [TODO: move after cbvalivw 1921]. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-modald 31848 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Theorembj-denot 31849* A weakening of ax-6 1875 and ax6v 1876. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)

Theorembj-eqs 31850* A lemma for substitutions, proved from Tarski's FOL. The version without DV(𝑥, 𝑦) is true but requires ax-13 2234. 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))

Theorembj-cbvexw 31851* Change bound variable. This is to cbvexvw 1957 what cbvalw 1955 is to cbvalvw 1956. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theorembj-ax12w 31852* The general statement that ax12w 1997 proves. (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))       (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))

21.14.4.8  Membership predicate, ax-8 and ax-9

Theorembj-elequ2g 31853* A form of elequ2 1991 with a universal quantifier. Its converse is ax-ext 2590. (TODO: move to main part, minimize axext4 2594--- as of 4-Nov-2020, minimizes only axext4 2594, by 13 bytes; and link to it in the comment of ax-ext 2590.) (Contributed by BJ, 3-Oct-2019.)
(𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-ax89 31854 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1979 and ax-9 1986. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1979 and ax-9 1986, as proved here. In the other direction, one can prove ax-8 1979 (respectively ax-9 1986) from bj-ax89 31854 by using mpan2 703 ( respectively mpan 702) and equid 1926. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-elequ12 31855 An identity law for the non-logical predicate, which combines elequ1 1984 and elequ2 1991. For the analogous theorems for class terms, see eleq1 2676, eleq2 2677 and eleq12 2678. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-cleljusti 31856* One direction of cleljust 1985, requiring only ax-1 6-- ax-5 1827 and ax8v1 1981. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)

Theorembj-alcomexcom 31857 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1728 section, soon after 2nexaln 1747, and used to prove excom 2029. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))

Theorembj-hbalt 31858 Closed form of hbal 2023. When in main part, prove hbal 2023 and hbald 2028 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremaxc11n11 31859 Proof of axc11n 2295 from { ax-1 6-- ax-7 1922, axc11 2302 } . Almost identical to axc11nfromc11 33229. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n11r 31860 Proof of axc11n 2295 from { ax-1 6-- ax-7 1922, axc9 2290, axc11r 2175 } (note that axc16 2120 is provable from { ax-1 6-- ax-7 1922, axc11r 2175 }).

Note that axc11n 2295 proves (over minimal calculus) that axc11 2302 and axc11r 2175 are equivalent. Therefore, axc11n11 31859 and axc11n11r 31860 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2302 appears slightly stronger since axc11n11r 31860 requires axc9 2290 while axc11n11 31859 does not).

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

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-axc16g16 31861* Proof of axc16g 2119 from { ax-1 6-- ax-7 1922, axc16 2120 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorembj-ax12v3 31862* A weak version of ax-12 2034 which is stronger than ax12v 2035. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 1926), then bj-ax12v3 31862 implies ax-5 1827 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 31863. (Contributed by BJ, 6-Jul-2021.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ax12v3ALT 31863* Alternate proof of bj-ax12v3 31862. Uses axc11r 2175 and axc15 2291 instead of ax-12 2034. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-sb 31864* A weak variant of sbid2 2401 not requiring ax-13 2234 nor ax-10 2006. On top of Tarski's FOL, one implication requires only ax12v 2035, and the other requires only sp 2041. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-modalbe 31865 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2127. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)

Theorembj-spst 31866 Closed form of sps 2043. Once in main part, prove sps 2043 and spsd 2045 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-19.21bit 31867 Closed form of 19.21bi 2047. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))

Theorembj-19.23bit 31868 Closed form of 19.23bi 2049. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))

Theorembj-nexrt 31869 Closed form of nexr 2050. Contrapositive of 19.8a 2039. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)

Theorembj-alrim 31870 Closed form of alrimi 2069. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-alrim2 31871 Imported form (uncurried form) of bj-alrim 31870. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))

Theorembj-nfdt0 31872 A theorem close to a closed form of nf5d 2104 and nf5dh 2013. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

Theorembj-nfdt 31873 Closed form of nf5d 2104 and nf5dh 2013. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))

Theorembj-nexdt 31874 Closed form of nexd 2076. (Contributed by BJ, 20-Oct-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))

Theorembj-nexdvt 31875* Closed form of nexdv 1851. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))

Theorembj-19.3t 31876 Closed form of 19.3 2057. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑𝜑))

Theorembj-alexbiex 31877 Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-exexbiex 31878 Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-alalbial 31879 Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∀𝑥𝜑)

Theorembj-exalbial 31880 Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∀𝑥𝜑)

Theorembj-19.9htbi 31881 Strengthening 19.9ht 2128 by replacing its succedent with a biconditional (19.9t 2059 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theorembj-hbntbi 31882 Strengthening hbnt 2129 by replacing its succedent with a biconditional. See also hbntg 30955 and hbntal 37790. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 31881. (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Theorembj-biexal1 31883 A general FOL biconditional that generalizes 19.9ht 2128 among others. For this and the following theorems, see also 19.35 1794, 19.21 2062, 19.23 2067. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexal2 31884 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexal3 31885 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))

Theorembj-bialal 31886 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexex 31887 A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))

Theorembj-hbext 31888 Closed form of hbex 2142. (Contributed by BJ, 10-Oct-2019.)
(∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Theorembj-nfalt 31889 Closed form of nfal 2139. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Theorembj-nfext 31890 Closed form of nfex 2140. (Contributed by BJ, 10-Oct-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Theorembj-eeanvw 31891* Version of eeanv 2170 with a DV condition on 𝑥, 𝑦 not requiring ax-11 2021. (The same can be done with eeeanv 2171 and ee4anv 2172.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theorembj-modal4e 31892 Dual statement of hba1 2137 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∃𝑥𝜑)

Theorembj-modalb 31893 A short form of the axiom B of modal logic. (Contributed by BJ, 4-Apr-2021.)
𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theorembj-axc10 31894 Alternate (shorter) proof of axc10 2240. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theorembj-alequex 31895 A fol lemma. Can be used to reduce the proof of spimt 2241 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Theorembj-spimt2 31896 A step in the proof of spimt 2241. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∃𝑥𝜓𝜓) → (∀𝑥𝜑𝜓)))

Theorembj-cbv3ta 31897 Closed form of cbv3 2253. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Theorembj-cbv3tb 31898 Closed form of cbv3 2253. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Theorembj-hbsb3t 31899 A theorem close to a closed form of hbsb3 2352. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theorembj-hbsb3 31900 Shorter proof of hbsb3 2352. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

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