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Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwl-nf2d 32301 Counterpart of nfd 2009 using nf2 2090 as a definition for 'not free'.

Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)

(𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremwl-nftht0 32302 Closed form of nfth 1707. (Contributed by BJ, 16-Sep-2021.)
(∀𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremwl-nfntht 32303 Closed form of nfnth 1710. (Contributed by BJ, 16-Sep-2021.)
(¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremwl-nfntht2 32304 Closed form of nfnth 1710.

[NOTE: theorems up to this point use only propositional calculus.]

(Contributed by BJ, 16-Sep-2021.)

(∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)

Theoremwl-nfth 32305 Revision of nfth 1707 using nf2 2090 as a definition for 'not free'. [ +- ]

No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.)

𝜑       𝑥𝜑

Theoremwl-nfnth 32306 Revision of nfnth 1710 using nf2 2090 as a definition for 'not free'. [ +- ]

No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.)

¬ 𝜑       𝑥𝜑

Theoremwl-nfbii 32307 Revision of nfbii 1728 using nf2 2090 as a definition for 'not free'. [ +- ]

Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.)

(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Theoremwl-nfxfr 32308 Revision of nfxfr 1729 using nf2 2090 as a definition for 'not free'. [ +- ]

A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑

Theoremwl-nfxfrd 32309 Revision of nfxfrd 1730 using nf2 2090 as a definition for 'not free'. [ +- ]

A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)

(𝜑𝜓)    &   (𝜒 → Ⅎ𝑥𝜓)       (𝜒 → Ⅎ𝑥𝜑)

Theoremwl-nf-nf2 32310 By ax-10 1966 the definition df-nf 1699 is stricter than the new nf2 2090 style. (Contributed by Wolf Lammen, 14-Sep-2021.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)       (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))

Theoremwl-nf2-nf 32311 hba1 2026 is sufficient to let a nf2 2090 style definition be stricter than df-nf 1699. (Contributed by Wolf Lammen, 14-Sep-2021.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)       ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))

Theoremwl-nfnt 32312 Revision of nfnt 2030 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Theoremwl-nfn 32313 Revision of nfn 2031 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

Inference associated with nfnt 2030. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

𝑥𝜑       𝑥 ¬ 𝜑

Theoremwl-nfnd 32314 Revision of nfnd 2032 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

Deduction associated with nfnt 2030. (Contributed by Mario Carneiro, 24-Sep-2016.)

(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)

Theoremwl-nfimd 32315 Revision of nfimd 2048 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremwl-nfim 32316 Revision of nfim 2051 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) Definition change. (Revised by Wolf Lammen, 17-Sep-2021.)

𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremwl-nfand 32317 Revision of nfand 2056 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)

(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremwl-nf3and 32318 Revision of nf3and 2057 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

Deduction form of bound-variable hypothesis builder nf3an 2061. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)

(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))

Theoremwl-nfan 32319 Revision of nfan 2059 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremwl-nfnan 32320 Revision of nfnan 2060 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by Scott Fenton, 2-Jan-2018.)

𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremwl-nf3an 32321 Revision of nf3an 2061 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)

𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)

Theoremwl-nfbid 32322 Revision of nfbid 2064 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremwl-nfbi 32323 Revision of nfbi 2065 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremwl-nfor 32324 Revision of nfor 2066 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.)

𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremwl-nf3or 32325 Revision of nf3or 2067 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ]

If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)

𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)

Theoremwl-nfv 32326* Revision of nfv 1796 using nf2 2090 as a definition for 'not free'. [ -ax-gen ]

If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.)

𝑥𝜑

Theoremwl-axc7ea 32327 Dual statement of hbe1 1968. [Modified version of axc7e 1993 with a universally quantified consequent.] (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Theoremwl-dfnf-1 32328 One direction of wl-dfnf 32351 can be proved with a smaller footprint on axiom usage. [ +ax-mp +ax-1 +ax-2 +ax-3 +ax-gen +ax-4 +ax-10 ] (Contributed by Wolf Lammen, 16-Sep-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Theoremwl-nfi 32329 Revision of nfi 1705 using nf2 2090 as a definition for 'not free'. [ +ax-4 +ax-10 ]

Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

(𝜑 → ∀𝑥𝜑)       𝑥𝜑

Theoremwl-nfe1 32330 Revision of nfe1 1969 using nf2 2090 as a definition for 'not free'. [ +- ]

The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)

𝑥𝑥𝜑

Theoremwl-nfdv 32331* Revision of nfdv 1818 using nf2 2090 as a definition for 'not free'. [ +ax-10 ]

Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremwl-nfdh 32332 Revision of nfdh 2010 using nf2 2090 as a definition for 'not free'. [ -ax-5 -ax-6 -ax-7 +ax-10 -ax-12 ]

Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) Definition change. (Revised by Wolf Lammen, 15-Sep-2021.)

(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremwl-nfr 32333 Revision of nfr 2004 using nf2 2090 as a definition for 'not free'. [ +- ]

Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Theoremwl-nfri 32334 Revision of nfri 2005 using nf2 2090 as a definition for 'not free'. [ +- ]

Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

𝑥𝜑       (𝜑 → ∀𝑥𝜑)

Theoremwl-nfrd 32335 Revision of nfrd 2006 using nf2 2090 as a definition for 'not free'. [ +- ]

Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))

Theoremwl-albid 32336 Revision of albid 2015 using nf2 2090 as a definition for 'not free'. [ +- ]

Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremwl-exbid 32337 Revision of exbid 2016 using nf2 2090 as a definition for 'not free'. [ +- ]

Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremwl-nfbidf 32338 Revision of exbid 2016 using nf2 2090 as a definition for 'not free'. [ +- ]

An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Theoremwl-19.9d 32339 Revision of 19.9d 2020 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

A deduction version of one direction of 19.9 2022. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))

Theoremwl-19.9t 32340 Revision of 19.9t 2021 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

A closed version of 19.9 2022. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)

(Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))

Theoremwl-19.21t 32341 Revision of 19.21t 2035 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2036. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theoremwl-19.21 32342 Revision of 19.21 2036 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 1821 for a version requiring fewer axioms. See also 19.21h 2038. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremwl-19.23t 32343 Revision of 19.23t 2040 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2041. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

(Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theoremwl-19.23 32344 Revision of 19.23 2041 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

Theorem 19.23 of [Margaris] p. 90. See 19.23v 1852 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremwl-alrimi 32345 Revision of alrimi 2008 using nf2 2090 as a definition for 'not free'. [ +- ]

Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2036. (Contributed by Mario Carneiro, 24-Sep-2016.)

𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremwl-exlimi 32346 Revision of exlimi 2043 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

Inference associated with 19.23 2041. See exlimiv 1811 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

𝑥𝜓    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremwl-nfim1 32347 Revision of nfim1 2050 using nf2 2090 as a definition for 'not free'. [ -ax-10 ]

A closed form of nfim 2051. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremwl-nfan1 32348 Revision of nfan1 2058 using nf2 2090 as a definition for 'not free'. [ +- ]

A closed form of nfan 2059. (Contributed by Mario Carneiro, 3-Oct-2016.) Definition change. (Revised by Wolf Lammen, 18-Sep-2021.)

𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremwl-nfa1 32349 Revision of nfa1 2027 using nf2 2090 as a definition for 'not free'. [ +- ]

The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

𝑥𝑥𝜑

Theoremwl-nfal 32350 Revision of nfal 2079 using nf2 2090 as a definition for 'not free'. [ +ax-10 ]

If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)

𝑥𝜑       𝑥𝑦𝜑

Theoremwl-dfnf 32351 Revision of df-nf 1699 using nf2 2090 as a definition for 'not free'. [ +ax-mp +ax-1 +ax-2 +ax-3 +ax-gen +ax-4 +ax-5 +ax-6 +ax-7 +ax-10 +ax-12 ]

The original definition of 'not free'. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 2272). An example of where this is used is stdpc5 2039. See nf2 2090 for an alternate definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, 𝑥 is effectively not free in the bare expression 𝑥 = 𝑥 (see nfequid 1890), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the expression 𝑥 = 𝑥 cannot affect the truth of the expression (and thus substitution will not change the result).

This predicate only applies to wffs. See df-nfc 2644 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.)

(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Theoremwl-nfd 32352 Revision of nfd 2009 using nf2 2090 as a definition for 'not free'. [ +ax-10 ]

Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremwl-nfdi 32353 Revision of nfdi 2047 using nf2 2090 as a definition for 'not free'. [ +ax-10 ]

Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either 𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)

(𝜑 → Ⅎ𝑥𝜑)       𝑥𝜑

20.17.4  An alternative axiom ~ ax-13

Axiomax-wl-13v 32354* A version of ax13v 2138 with a distinctor instead of a distinct variable expression.

Had we additonally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1793. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremwl-ax13lem1 32355* A version of ax-wl-13v 32354 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

20.17.5  Other stuff

Theoremwl-jarri 32356 Dropping a nested antecedent. This theorem is one of two reversions of ja 171. Since ja 171 is reversible, a nested (chain of) implication(s) is just a packed notation of two or more theorems/hypotheses with a common consequent. axc5c7 33104 is an instance of this idea. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theoremwl-jarli 32357 Dropping a nested consequent. This theorem is one of two reversions of ja 171. Since ja 171 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. axc5c7 33104 is an instance of this idea. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       𝜑𝜒)

Theoremwl-mps 32358 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremwl-syls1 32359 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremwl-syls2 32360 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremwl-embant 32361 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremwl-orel12 32362 In a conjunctive normal form a pair of nodes like (𝜑𝜓) ∧ (¬ 𝜑𝜒) eliminates the need of a node (𝜓𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
(((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → (𝜓𝜒))

Theoremwl-cases2-dnf 32363 A particular instance of orddi 908 and anddi 909 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1004, and is related to consensus 989. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1006 and dfifp4 1009, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Theoremwl-dfnan2 32364 An alternative definition of "nand" based on imnan 436. See df-nan 1439 for the original definition. This theorem allows various shortenings. (Contributed by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))

Theoremwl-nancom 32365 The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Revised by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremwl-nannan 32366 Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theoremwl-nannot 32367 Show equivalence between negation and the Nicod version. To derive nic-dfneg 1585, apply nanbi 1445. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
𝜑 ↔ (𝜑𝜑))

Theoremwl-nanbi1 32368 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremwl-nanbi2 32369 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremwl-naev 32370* If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Theoremwl-hbae1 32371 This specialization of hbae 2207 does not depend on ax-11 1971. (Contributed by Wolf Lammen, 8-Aug-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)

Theoremwl-naevhba1v 32372* An instance of hbn1w 1922 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-hbnaev 32373* Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 1966. (Contributed by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-spae 32374 Prove an instance of sp 1990 from ax-13 2137 and Tarski's FOL only, without distinct variable conditions. The antecedent 𝑥𝑥 = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for 𝑥, or vice versa, i.e. both variables are effectively the same. The converse ¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies 𝑥𝑥 = 𝑦 and ¬ ∀𝑥𝑥 = 𝑦 can help eliminating distinct variable conditions.

The antecedent 𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'.

Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 1983.

Note that this theorem is also provable from ax-12 1983 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1847 and spaev 1926 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremwl-cbv3vv 32375* Avoiding ax-11 1971. (Contributed by Wolf Lammen, 30-Aug-2021.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremwl-speqv 32376* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 1990 is provable from Tarski's FOL and ax13v 2138 only. Note that this reverts the implication in ax13lem1 2139, so in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))

Theoremwl-19.8eqv 32377* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 1988 is provable from Tarski's FOL and ax13v 2138 only. Note that this reverts the implication in ax13lem2 2187, so in fact 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦))

Theoremwl-19.2reqv 32378* Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1842 is provable from Tarski's FOL and ax13v 2138 only. Note that in conjunction with 19.2 1842 in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremwl-dveeq12 32379* The current form of ax-13 2137 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 1966 to arrive at the more general form presented here. You need 19.8a 1988 (or ax-12 1983) to restore 𝑦 = 𝑧 from 𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremwl-nfalv 32380* If 𝑥 is not present in 𝜑, it is not free in 𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.)
𝑥𝑦𝜑

Theoremwl-nfimf1 32381 An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 2051 in dvelimdf 2227 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
(∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))

Theoremwl-nfnbi 32382 Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 2031 or nfnd 2032. (Contributed by Wolf Lammen, 5-May-2018.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Theoremwl-nfae1 32383 Unlike nfae 2208, this specialized theorem avoids ax-11 1971. (Contributed by Wolf Lammen, 26-Jun-2019.)
𝑥𝑦 𝑦 = 𝑥

Theoremwl-nfnae1 32384 Unlike nfnae 2210, this specialized theorem avoids ax-11 1971. (Contributed by Wolf Lammen, 27-Jun-2019.)
𝑥 ¬ ∀𝑦 𝑦 = 𝑥

Theoremwl-aetr 32385 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))

Theoremwl-dral1d 32386 A version of dral1 2217 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, 𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 32398 and nfdi 2047 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))

Theoremwl-cbvalnaed 32387 wl-cbvalnae 32388 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremwl-cbvalnae 32388 A more general version of cbval 2162 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2226, nfsb2 2252 or dveeq1 2192. (Contributed by Wolf Lammen, 4-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremwl-exeq 32389 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

Theoremwl-aleq 32390 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))

Theoremwl-nftht 32391 A tautological version of nfth 1707. This convenience theorem allows easy replacement of 𝑥𝜑 antecedents with 𝑥𝜑, for example in 19.9t 2021. (Contributed by Wolf Lammen, 19-Aug-2018.)
(∀𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremwl-nfeqfb 32392 Extend nfeqf 2193 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
(Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))

Theoremwl-nfs1t 32393 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2257. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremwl-equsald 32394 Deduction version of equsal 2182. (Contributed by Wolf Lammen, 27-Jul-2019.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Theoremwl-equsal 32395 A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 32394 first, and then deriving more specialized versions wl-equsal 32395 and wl-equsal1t 32396 then is more efficient than the other way round, which is possible, too. See also equsal 2182. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremwl-equsal1t 32396 The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑.

This theorem is more fundamental than equsal 2182, spimt 2144 or sbft 2271, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)

(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Theoremwl-equsalcom 32397 This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑦 = 𝑥𝜑))

Theoremwl-equsal1i 32398 The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
(Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)    &   (𝑥 = 𝑦𝜑)       𝜑

Theoremwl-sb6rft 32399 A specialization of wl-equsal1t 32396. Closed form of sb6rf 2315. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))

Theoremwl-sbrimt 32400 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2288. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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