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Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-nftru 31601 The true constant has no free variables. (Contributed by BJ, 6-May-2019.)
ℲℲ𝑥

Theorembj-nfnth 31602 Any variable is not free in a falsity. (Contributed by BJ, 6-May-2019.)
¬ 𝜑       ℲℲ𝑥𝜑

Theorembj-nffal 31603 The false constant has no free variables. (Contributed by BJ, 6-May-2019.)
ℲℲ𝑥

Theorembj-genr 31604 Generalization rule on the right conjunct. See 19.28 2055. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (𝜑 ∧ ∀𝑥𝜓)

Theorembj-genl 31605 Generalization rule on the left conjunct. See 19.27 2054. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorembj-genan 31606 Generalization rule on a conjunction. Forward inference associated with 19.26 1767. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Theorembj-2alim 31607 Closed form of 2alimi 1716. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))

Theorembj-2exim 31608 Closed form of 2eximi 1741. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))

Theorembj-alanim 31609 Closed form of alanimi 1719. (Contributed by BJ, 6-May-2019.)
(∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Theorembj-2albi 31610 Closed form of 2albii 1723. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))

Theorembj-notalbii 31611 Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 3808, ballotlem2 29678, bnj1143 29954, hausdiag 21164. (Contributed by BJ, 17-Jul-2021.)
(𝜑𝜓)       (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Theorembj-2exbi 31612 Closed form of 2exbii 1753. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Theorembj-3exbi 31613 Closed form of 3exbii 1754. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Theorembj-sylgt2 31614 Uncurried form of sylgt 1724. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜓𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))

Theorembj-exlimh 31615 Closed form of close to exlimih 2044. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑𝜓) → ((∃𝑥𝜓𝜒) → (∃𝑥𝜑𝜒)))

Theorembj-exlimh2 31616 Uncurried form of bj-exlimh 31615. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜑𝜓) ∧ (∃𝑥𝜓𝜒)) → (∃𝑥𝜑𝜒))

Theorembj-alrimhi 31617 An inference associated with sylgt 1724 and bj-exlimh 31615. (Contributed by BJ, 12-May-2019.)
(𝜑𝜓)       (ℲℲ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-nexdh 31618 Closed form of nexdh 1760 (and more general since it uses 𝜒). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Theorembj-nexdh2 31619 Uncurried form of bj-nexdh 31618. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))

Theorembj-hbxfrbi 31620 Closed form of hbxfrbi 1727. Notes: it is less important than bj-nfbi 31621; it requires sp 1990 (unlike bj-nfbi 31621); there is an obvious version with (∃𝑥𝜑𝜑) instead. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Theorembj-nfbi 31621 Closed form of nfbii 1728 (with df-bj-nf 31593 instead of df-nf 1699, which would require more axioms). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓))

Theorembj-nfxfr 31622 Proof of nfxfr 1729 from bj-nfbi 31621. (Contributed by BJ, 6-May-2019.)
(𝜑𝜓)    &   ℲℲ𝑥𝜑       ℲℲ𝑥𝜓

Theorembj-nfn 31623 A variable is non-free in a proposition if and only if it is so in its negation. Requires fewer axioms than nfn 2031. (Contributed by BJ, 6-May-2019.)
(ℲℲ𝑥𝜑 ↔ ℲℲ𝑥 ¬ 𝜑)

Theorembj-exlime 31624 Variant of exlimih 2044 where the non-freeness of 𝑥 in 𝜓 is expressed using an existential quantifier. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝜓𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theorembj-exnalimn 31625 A transformation of quantifiers and logical connectives. The general statement that equs3 1825 proves.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1793. I propose to move to the main part: bj-exnalimn 31625, bj-exaleximi 31628, bj-exalimi 31629, bj-ax12i 31631, bj-ax12wlem 31635, bj-ax12w 31680, and remove equs3 1825. A new label is needed for bj-ax12i 31631 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 1826 and spimfw 1828 (other spim* theorems use 𝑥 and very very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 29-Sep-2019.)

(∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Theorembj-nalnaleximiOLD 31626 An inference for distributing quantifiers over a double implication. The general statement that speimfw 1826 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))

Theorembj-nalnalimiOLD 31627 An inference for distributing quantifiers over a double implication. The general statement that spimfw 1828 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝜓 → ∀𝑥 ¬ 𝜓)       (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑𝜓))

Theorembj-exaleximi 31628 An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1826 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Theorembj-exalimi 31629 An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1828 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Theorembj-ax12ig 31630 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 31631. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12i 31631 A weakening of bj-ax12ig 31630 that is sufficient to prove a weak form of the axiom of substitution ax-12 1983. The general statement of which ax12i 1829 is an instance. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12iOLD 31632 Old proof of bj-ax12i 31631. 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 31633* 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 31634* A non-occurring variable is semantically non-free. (Contributed by BJ, 6-May-2019.)
ℲℲ𝑥𝜑

Theorembj-ax12wlem 31635* A lemma used to prove a weak version of the axiom of substitution ax-12 1983. (Temporary comment: The general statement that ax12wlem 1957 proves.) (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

20.14.4.5  Equality and substitution

Syntaxwssb 31636 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 31637* 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 2321 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2347 with respect to sb5 2322.

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 1831 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 1990, of the substitution axiom ax-12 1983, and of commutation of quantifiers ax-11 1971; that is, ax-13 2137 will often be avoided. (Contributed by BJ, 22-Dec-2020.)

([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbim 31638 Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2286. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Theorembj-ssbbi 31639 Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2294. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Theorembj-ssbimi 31640 Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)

Theorembj-ssbbii 31641 Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)

Theorembj-alsb 31642 If a proposition is true for all instances, then it is true for any specific one. See stdpc4 2245. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-sbex 31643 If a proposition is true for a specific instance, then there exists an instance such that it is true for it. See spsbe 1834. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)

Theorembj-ssbeq 31644* 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 31645* Substitution for a variable not occurring in a proposition. See sbf 2272. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑𝜑)

Theorembj-ssbequ 31646 Equality property for substitution, from Tarski's system. Compare sbequ 2268. (Contributed by BJ, 30-Dec-2020.)
(𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑))

Theorembj-ssblem1 31647* A lemma for the definiens of df-sb 1831. (Contributed by BJ, 22-Dec-2020.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssblem2 31648* The converse may not be provable without ax-11 1971. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))

Theorembj-ssb1a 31649* One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 31650 for the biconditional, which requires ax-11 1971. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)

Theorembj-ssb1 31650* A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 31649 for the backward implication, which does not require ax-11 1971 (note that here, the version of ax-11 1971 with disjoint setvar metavariables would suffice). Compare sb6 2321. (Contributed by BJ, 22-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))

Theorembj-ax12 31651* A weaker form of ax-12 1983 and ax12v2 1985, 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 31652* The axiom bj-ax12 31651 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
[𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)

Theorembj-modal5e 31653 Dual statement of hbe1 1968 (which is the real modal-5 1970). See also axc7 1992 and axc7e 1993. (Contributed by BJ, 21-Dec-2020.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Theorembj-19.41al 31654 Special case of 19.41 2101 proved from Tarski, ax-10 1966 (modal5) and hba1 2026 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Theorembj-equsexval 31655* Special case of equsexv 2117 proved from Tarski, ax-10 1966 (modal5) and hba1 2026 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)

Theorembj-sb56 31656* Proof of sb56 2128 from Tarski, ax-10 1966 (modal5) and bj-ax12 31651. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-dfssb2 31657* An alternate definition of df-ssb 31637. Note that the use of a dummy variable in the definition df-ssb 31637 allows to use bj-sb56 31656 instead of equs45f 2242 and hence to avoid dependency on ax-13 2137 and to use ax-12 1983 only through bj-ax12 31651. Compare dfsb7 2347. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbn 31658 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 1966, bj-ax12 31651. Compare sbn 2283. (Contributed by BJ, 25-Dec-2020.)
([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑)

Theorembj-ssbft 31659 See sbft 2271. This proof is from Tarski's FOL together with sp 1990 (and its dual). (Contributed by BJ, 22-Dec-2020.)
(ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ2 31660 Note that ax-12 1983 is used only via sp 1990. See sbequ2 1832 and stdpc7 1908. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))

Theorembj-ssbequ1 31661 This uses ax-12 1983 with a direct reference to ax12v 1984. Therefore, compared to bj-ax12 31651, there is a hidden use of sp 1990. Note that with ax-12 1983, it can be proved with dv condition on 𝑥, 𝑡. See sbequ1 2129. (Contributed by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))

Theorembj-ssbid2 31662 A special case of bj-ssbequ2 31660. (Contributed by BJ, 22-Dec-2020.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid2ALT 31663 Alternate proof of bj-ssbid2 31662, not using bj-ssbequ2 31660. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥/𝑥]b𝜑𝜑)

Theorembj-ssbid1 31664 A special case of bj-ssbequ1 31661. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbid1ALT 31665 Alternate proof of bj-ssbid1 31664, not using bj-ssbequ1 31661. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥/𝑥]b𝜑)

Theorembj-ssbssblem 31666* 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 31667* 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 31668* Lemma for bj-ax6e 31670. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theorembj-ax6elem2 31669* Lemma for bj-ax6e 31670. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Theorembj-ax6e 31670 Proof of ax6e 2141 (hence ax6 2142) from Tarski's system, ax-c9 33077, ax-c16 33079. Remark: ax-6 1838 is used only via its principal (unbundled) instance ax6v 1839. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theorembj-extru 31671 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 1842. (This is also extt 31401; propose to move to Main extt 31401 and allt 31398; relabel exiftru 1841 to exgen ? ). (Contributed by BJ, 12-May-2019.) (Proof modification is discouraged.)
𝑥

Theorembj-spimevw 31672* Existential introduction, using implicit substitution. This is to spimeh 1875 what spimvw 1877 is to spimw 1876. (Contributed by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spnfw 31673 Theorem close to a closed form of spnfw 1878. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-cbvexiw 31674* Change bound variable. This is to cbvexvw 1919 what cbvaliw 1883 is to cbvalvw 1918. [TODO: move after cbvalivw 1884]. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbvexivw 31675* Change bound variable. This is to cbvexvw 1919 what cbvalivw 1884 is to cbvalvw 1918. [TODO: move after cbvalivw 1884]. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-modald 31676 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Theorembj-denot 31677* A weakening of ax-6 1838 and ax6v 1839. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)

Theorembj-eqs 31678* A lemma for substitutions, proved from Tarski's FOL. The version without DV(𝑥, 𝑦) is true but requires ax-13 2137. 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 31679* Change bound variable. This is to cbvexvw 1919 what cbvalw 1917 is to cbvalvw 1918. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theorembj-ax12w 31680* The general statement that ax12w 1958 proves. (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))       (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))

20.14.4.8  Membership predicate, ax-8 and ax-9

Theorembj-elequ2g 31681* A form of elequ2 1952 with a universal quantifier. Its converse is ax-ext 2494. (TODO: move to main part, minimize axext4 2498--- as of 4-Nov-2020, minimizes only axext4 2498, by 13 bytes; and link to it in the comment of ax-ext 2494.) (Contributed by BJ, 3-Oct-2019.)
(𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-ax89 31682 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1940 and ax-9 1947. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1940 and ax-9 1947, as proved here. In the other direction, one can prove ax-8 1940 (respectively ax-9 1947) from bj-ax89 31682 by using mpan2 702 ( respectively mpan 701) and equid 1889. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-elequ12 31683 An identity law for the non-logical predicate, which combines elequ1 1945 and elequ2 1952. For the analogous theorems for class terms, see eleq1 2580, eleq2 2581 and eleq12 2582. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-cleljusti 31684* One direction of cleljust 1946, requiring only ax-1 6-- ax-5 1793 and ax8v1 1942. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)

Theorembj-alcomexcom 31685 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1713 section, soon after 2nexaln 1735, and used to prove excom 1978. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))

Theorembj-hbalt 31686 Closed form of hbal 1973. When in main part, prove hbal 1973 and hbald 1977 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremaxc11n11 31687 Proof of axc11n 2199 from { ax-1 6-- ax-7 1885, axc11 2206 } . Almost identical to axc11nfromc11 33113. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n11r 31688 Proof of axc11n 2199 from { ax-1 6-- ax-7 1885, axc9 2194, axc11r 2136 } (note that axc16 2072 is provable from { ax-1 6-- ax-7 1885, axc11r 2136 }).

Note that axc11n 2199 proves (over minimal calculus) that axc11 2206 and axc11r 2136 are equivalent. Therefore, axc11n11 31687 and axc11n11r 31688 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2206 appears slightly stronger since axc11n11r 31688 requires axc9 2194 while axc11n11 31687 does not).

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

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-axc16g16 31689* Proof of axc16g 2071 from { ax-1 6-- ax-7 1885, axc16 2072 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorembj-ax12v3 31690* A weak version of ax-12 1983 which is stronger than ax12v 1984. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 1889), then bj-ax12v3 31690 implies ax-5 1793 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 31691. (Contributed by BJ, 6-Jul-2021.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ax12v3ALT 31691* Alternate proof of bj-ax12v3 31690. Uses axc11r 2136 and axc15 2195 instead of ax-12 1983. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-sb 31692* A weak variant of sbid2 2305 not requiring ax-13 2137 nor ax-10 1966. On top of Tarski's FOL, one implication requires only ax12v 1984, and the other requires only sp 1990. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-modalbe 31693 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 1994. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)

Theorembj-spst 31694 Closed form of sps 1996. Once in main part, prove sps 1996 and spsd 1998 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-19.21bit 31695 Closed form of 19.21bi 2000. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))

Theorembj-19.23bit 31696 Closed form of 19.23bi 2002. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))

Theorembj-nexrt 31697 Closed form of nexr 2003. Contrapositive of 19.8a 1988. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)

Theorembj-alrim 31698 Closed form of alrimi 2008. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-alrim2 31699 Imported form (uncurried form) of bj-alrim 31698. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))

Theorembj-nfdt0 31700 A theorem close to a closed form of nfd 2009 and nfdh 2010. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

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