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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreximdv2 3201* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
(𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
 
Theoremreximdvai 3202* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
TheoremreximdvaiOLD 3203* Obsolete version of reximdvai 3202 as of 3-Nov-2024. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximdv 3204* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximdva 3205* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 
Theoremreximddv 3206* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremreximssdv 3207* Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)
(𝜑 → ∃𝑥𝐵 𝜓)    &   ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝑥𝐴)    &   ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝜒)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremreximdvva 3208* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremreximddv2 3209* Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
 
Theoremr19.23v 3210* Restricted quantifier version of 19.23v 1949. Version of r19.23 3245 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremrexlimiv 3211* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimiva 3212* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((𝑥𝐴𝜑) → 𝜓)       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimivw 3213* Weaker version of rexlimiv 3211. (Contributed by FL, 19-Sep-2011.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimdv 3214* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
(𝜑 → (𝑥𝐴 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdva 3215* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdvaa 3216* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdv3a 3217* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 3214. (Contributed by NM, 7-Jun-2015.)
((𝜑𝑥𝐴𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimdva2 3218* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremr19.29an 3219* A commonly used pattern in the spirit of r19.29 3186. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
 
Theoremr19.29a 3220* A commonly used pattern in the spirit of r19.29 3186. (Contributed by Thierry Arnoux, 22-Nov-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theoremrexlimdvw 3221* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimddv 3222* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)       (𝜑𝜒)
 
Theoremrexlimivv 3223* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
 
Theoremrexlimdvv 3224* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
(𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremrexlimdvva 3225* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 
Theoremrexbidv2 3226* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremrexbidva 3227* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Dec-2019.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremrexbidv 3228* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theorem2rexbiia 3229* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
 
Theorem2rexbidva 3230* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theorem2rexbidv 3231* Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremrexralbidv 3232* Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 
Theoremr2exlem 3233 Lemma factoring out common proof steps in r2exf 3257 an r2ex 3234. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 10-Jan-2020.)
(∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremr2ex 3234* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremrspe 3235 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((𝑥𝐴𝜑) → ∃𝑥𝐴 𝜑)
 
Theoremrsp2e 3236 Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremnfre1 3237 The setvar 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝑥𝐴 𝜑
 
Theoremnfrexd 3238* Deduction version of nfrex 3240. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2374. See nfrexdg 3239 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfrexdg 3239 Deduction version of nfrexg 3241. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfrexd 3238 for a version with a disjoint variable condition, but not requiring ax-13 2374. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
Theoremnfrex 3240* 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 2374. See nfrexg 3241 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
Theoremnfrexg 3241 Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfrex 3240 for a version with a disjoint variable condition, but not requiring ax-13 2374. (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 3242 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 3243 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 3244 Closed theorem form of r19.23 3245. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 
Theoremr19.23 3245 Restricted quantifier version of 19.23 2208. See r19.23v 3210 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 3246 Restricted quantifier version of exlimi 2214. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑥𝜓    &   (𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑𝜓)
 
Theoremrexlimd2 3247 Version of rexlimd 3248 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 3248 Deduction form of rexlimd 3248. (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 3249 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
TheoremrexbidvaALT 3250* Alternate proof of rexbidva 3227, 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 3251 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 
TheoremrexbidvALT 3252* Alternate proof of rexbidv 3228, 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 3253 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.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 
TheoremralrexbidOLD 3254 Obsolete version of ralrexbid 3253 as of 4-Nov-2024. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 
Theoremr19.12 3255* Restricted quantifier version of 19.12 2325. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2374, ax-ext 2711. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremr19.12OLD 3256* Obsolete version of 19.12 2325 as of 4-Nov-2024. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2374, ax-ext 2711. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremr2exf 3257* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3233. (Revised by Wolf Lammen, 10-Jan-2020.)
𝑦𝐴       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 
Theoremrexeqbii 3258 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 3259 Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.)
(∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥𝐴 𝜑)
 
Theoremrmoanid 3260 Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.)
(∃*𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃*𝑥𝐴 𝜑)
 
Theoremr19.29af2 3261 A commonly used pattern based on r19.29 3186. (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 3262* A commonly used pattern based on r19.29 3186. See r19.29a 3220, r19.29an 3219 for a variant when 𝑥 is disjoint from 𝜑. (Contributed by Thierry Arnoux, 29-Nov-2017.)
𝑥𝜑    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜒)
 
Theorem2r19.29 3263 Theorem r19.29 3186 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)
((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
 
Theoremr19.29d2r 3264 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)       (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
 
Theoremr19.29d2rOLD 3265 Obsolete version of r19.29d2r 3264 as of 4-Nov-2024. (Contributed by Thierry Arnoux, 30-Jan-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)       (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
 
Theoremr19.29vva 3266* A commonly used pattern based on r19.29 3186, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)
 
Theoremr19.29vvaOLD 3267* Obsolete version of r19.29vva 3266 as of 4-Nov-2024. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 29-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)
 
Theoremr19.30 3268 Restricted quantifier version of 19.30 1888. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 5-Nov-2024.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.30OLD 3269 Obsolete version of 19.30 1888 as of 5-Nov-2024. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.32v 3270* Restricted quantifier version of 19.32v 1947. (Contributed by NM, 25-Nov-2003.)
(∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
 
Theoremr19.35 3271 Restricted quantifier version of 19.35 1884. (Contributed by NM, 20-Sep-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.36v 3272* Restricted quantifier version of one direction of 19.36 2227. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4443.) (Contributed by NM, 22-Oct-2003.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
 
Theoremr19.37 3273 Restricted quantifier version of one direction of 19.37 2229. (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 3274* Restricted quantifier version of one direction of 19.37v 1999. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4438.) (Contributed by NM, 2-Apr-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Jun-2023.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 
Theoremr19.40 3275 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 3276* Restricted quantifier version 19.41v 1957. Version of r19.41 3277 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 3277 Restricted quantifier version of 19.41 2232. See r19.41v 3276 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.41vv 3278* Version of r19.41v 3276 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
 
Theoremr19.42v 3279* Restricted quantifier version of 19.42v 1961 (see also 19.42 2233). (Contributed by NM, 27-May-1998.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))
 
Theoremr19.43 3280 Restricted quantifier version of 19.43 1889. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremr19.44v 3281* One direction of a restricted quantifier version of 19.44 2234. The other direction holds when 𝐴 is nonempty, see r19.44zv 4440. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
 
Theoremr19.45v 3282* Restricted quantifier version of one direction of 19.45 2235. The other direction holds when 𝐴 is nonempty, see r19.45zv 4439. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
 
Theoremralcom 3283* Commutation of restricted universal quantifiers. See ralcom2 3291 for a version without disjoint variable condition on 𝑥, 𝑦. This theorem should be used in place of ralcom2 3291 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 3284* 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)
 
Theoremralcomf 3285* Commutation of restricted universal quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcomf 3286* Commutation of restricted existential quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralcom13 3287* Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremrexcom13 3288* Swap first and third restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 
Theoremralrot3 3289* Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
(∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
 
Theoremrexrot4 3290* Rotate four restricted existential quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
 
Theoremralcom2 3291* 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 2711 and it should no longer be used. Usage of ralcom 3283 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 3292 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 3293* Lemma factoring out common proof steps of reeanv 3295 and reean 3294. (Contributed by Wolf Lammen, 20-Aug-2023.)
(∃𝑥𝑦((𝑥𝐴𝜑) ∧ (𝑦𝐵𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦(𝑦𝐵𝜓)))       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theoremreean 3294* 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 3295* Rearrange restricted existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 
Theorem3reeanv 3296* Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
 
Theorem2ralor 3297* Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Wolf Lammen, 20-Nov-2024.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
 
Theorem2ralorOLD 3298* Obsolete version of 2ralor 3297 as of 20-Nov-2024. (Contributed by Jeff Madsen, 19-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
 
Theoremnfreu1 3299 The setvar 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑
 
Theoremnfrmo1 3300 The setvar 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑
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