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Theorem List for Metamath Proof Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfrex 3301* Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) Add disjoint variable condition to avoid ax-13 2392. See nfrexg 3302 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfrexg 3302 Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2392. See nfrex 3301 for a version with a disjoint variable condition, but not requiring ax-13 2392. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremreximdai 3303 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximd2a 3304 Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥𝐵)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐵 𝜒)
 
Theoremr19.23t 3305 Closed theorem form of r19.23 3306. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 
Theoremr19.23 3306 Restricted quantifier version of 19.23 2213. See r19.23v 3271 for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
𝑥𝜓       (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremrexlimi 3307 Restricted quantifier version of exlimi 2219. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜓    &   (𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimd2 3308 Version of rexlimd 3309 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimd 3309 Deduction form of rexlimd 3309. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 14-Jan-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexbida 3310 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
TheoremrexbidvaALT 3311* Alternate proof of rexbidva 3288, shorter but requires more axioms. (Contributed by NM, 9-Mar-1997.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremrexbid 3312 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
TheoremrexbidvALT 3313* Alternate proof of rexbidv 3289, shorter but requires more axioms. (Contributed by NM, 20-Nov-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremralrexbid 3314 Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 
TheoremralrexbidOLD 3315 Obsolete version of ralrexbid 3314 as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 
Theoremr19.12 3316* Restricted quantifier version of 19.12 2348. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2392, ax-ext 2796. (Revised by Wolf Lammen, 17-Jun-2023.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremr2exf 3317* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3294. (Revised by Wolf Lammen, 10-Jan-2020.)
𝑦𝐴       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremrexeqbii 3318 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
 
Theoremreuanid 3319 Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.)
(∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥𝐴 𝜑)
 
Theoremrmoanid 3320 Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.)
(∃*𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃*𝑥𝐴 𝜑)
 
Theoremr19.29af2 3321 A commonly used pattern based on r19.29 3248. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝑥𝜑    &   𝑥𝜒    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29af 3322* A commonly used pattern based on r19.29 3248. See r19.29a 3281, r19.29an 3280 for a variant when 𝑥 is disjoint from 𝜑. (Contributed by Thierry Arnoux, 29-Nov-2017.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremr19.29anOLD 3323* Obsolete version of r19.29an 3280 as of 17-Jun-2023. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
 
Theoremr19.29aOLD 3324* Obsolete proof of r19.29a 3281 as of 17-Jun-2023. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theorem2r19.29 3325 Theorem r19.29 3248 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)
((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
 
Theoremr19.29d2r 3326 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)       (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
 
Theoremr19.29vva 3327* A commonly used pattern based on r19.29 3248, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)
 
Theoremr19.29vvaOLD 3328* Obsolete version of r19.29vva 3327 as of 28-Jun-2023. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)
 
Theoremr19.30 3329 Restricted quantifier version of 19.30 1883. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.30OLD 3330 Obsolete version of r19.30 3329 as of 18-Jun-2023. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.32v 3331* Restricted quantifier version of 19.32v 1942. (Contributed by NM, 25-Nov-2003.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
 
Theoremr19.35 3332 Restricted quantifier version of 19.35 1879. (Contributed by NM, 20-Sep-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.36v 3333* Restricted quantifier version of one direction of 19.36 2234. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4435.) (Contributed by NM, 22-Oct-2003.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
 
Theoremr19.37 3334 Restricted quantifier version of one direction of 19.37 2236. (The other direction does not hold when 𝐴 is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑       (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.37v 3335* Restricted quantifier version of one direction of 19.37v 1999. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4430.) (Contributed by NM, 2-Apr-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Jun-2023.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.37vOLD 3336* Obsolete version of r19.37v 3335 as of 18-Jun-2023. (Contributed by NM, 2-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.40 3337 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.41v 3338* Restricted quantifier version 19.41v 1951. Version of r19.41 3339 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 17-Dec-2003.) Reduce dependencies on axioms. (Revised by BJ, 29-Mar-2020.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.41 3339 Restricted quantifier version of 19.41 2239. See r19.41v 3338 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.41vv 3340* Version of r19.41v 3338 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
 
Theoremr19.42v 3341* Restricted quantifier version of 19.42v 1955 (see also 19.42 2240). (Contributed by NM, 27-May-1998.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.43 3342 Restricted quantifier version of 19.43 1884. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.44v 3343* One direction of a restricted quantifier version of 19.44 2241. The other direction holds when 𝐴 is nonempty, see r19.44zv 4432. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.45v 3344* Restricted quantifier version of one direction of 19.45 2242. The other direction holds when 𝐴 is nonempty, see r19.45zv 4431. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremralcom 3345* Commutation of restricted universal quantifiers. See ralcom2 3354 for a version without disjoint variable condition on 𝑥, 𝑦. This theorem should be used in place of ralcom2 3354 since it depends on a smaller set of axioms. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom 3346* Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
TheoremrexcomOLD 3347* Obsolete version of rexcom 3346 as of 26-Aug-2023. Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralcomf 3348* Commutation of restricted universal quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcomf 3349* Commutation of restricted existential quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralcom13 3350* Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom13 3351* Swap first and third restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralrot3 3352* Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
 
Theoremrexrot4 3353* Rotate four restricted existential quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
 
Theoremralcom2 3354* Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). This theorem relies on the full set of axioms up to ax-ext 2796 and it should no longer be used. Usage of ralcom 3345 is highly encouraged. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (New usage is discouraged.)
(∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑)
 
Theoremralcom3 3355 A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
 
Theoremreeanlem 3356* Lemma factoring out common proof steps of reeanv 3358 and reean 3357. (Contributed by Wolf Lammen, 20-Aug-2023.)
(∃𝑥𝑦((𝑥𝐴𝜑) ∧ (𝑦𝐵𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦(𝑦𝐵𝜓)))       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theoremreean 3357* Rearrange restricted existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theoremreeanv 3358* Rearrange restricted existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theorem3reeanv 3359* Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
 
Theorem2ralor 3360* Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
 
Theoremnfreu1 3361 The setvar 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑
 
Theoremnfrmo1 3362 The setvar 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑
 
Theoremnfreud 3363 Deduction version of nfreu 3367. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
 
Theoremnfrmod 3364 Deduction version of nfrmo 3368. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)
 
Theoremnfreuw 3365* Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3367 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑
 
Theoremnfrmow 3366* Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3368 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥∃*𝑦𝐴 𝜑
 
Theoremnfreu 3367 Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker nfreuw 3365 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑
 
Theoremnfrmo 3368 Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker nfrmow 3366 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜑       𝑥∃*𝑦𝐴 𝜑
 
Theoremrabid 3369 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
(𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
 
Theoremrabrab 3370 Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.)
{𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}
 
Theoremrabidim1 3371 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
 
Theoremrabid2 3372* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
 
Theoremrabid2f 3373 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
𝑥𝐴       (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
 
Theoremrabbi 3374 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3463. (Contributed by NM, 25-Nov-2013.)
(∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremnfrab1 3375 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
𝑥{𝑥𝐴𝜑}
 
Theoremnfrabw 3376* A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3377 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝜑    &   𝑥𝐴       𝑥{𝑦𝐴𝜑}
 
Theoremnfrab 3377 A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker nfrabw 3376 when possible. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝐴       𝑥{𝑦𝐴𝜑}
 
Theoremreubida 3378 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubidva 3379* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubidv 3380* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubiia 3381 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
(𝑥𝐴 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 
Theoremreubii 3382 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)
(𝜑𝜓)       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 
Theoremrmobida 3383 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobidva 3384* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobidv 3385* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobiia 3386 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 
Theoremrmobii 3387 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 
Theoremraleqf 3388 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 
Theoremrexeqf 3389 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 
Theoremreueq1f 3390 Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 
Theoremrmoeq1f 3391 Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
 
Theoremraleqbidv 3392* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexeqbidv 3393* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremraleqbi1dv 3394* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (Proof shortened by Steven Nguyen, 5-May-2023.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
 
Theoremrexeqbi1dv 3395* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (Proof shortened by Steven Nguyen, 5-May-2023.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))
 
Theoremraleq 3396* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 
Theoremrexeq 3397* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 
Theoremreueq1 3398* Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 
Theoremrmoeq1 3399* Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
 
Theoremraleqi 3400* Equality inference for restricted universal quantifier. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)
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