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Type | Label | Description |
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Statement | ||
Theorem | cbvreuv 3401* | Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3398 for a version without ax-13 2379, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvreuvw 3398 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrmov 3402* | Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvraldva2 3403* | Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvrexdva2 3404* | Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2023.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvrexdva2OLD 3405* | Obsolete version of cbvrexdva 3407 as of 12-Aug-2023. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvraldva 3406* | Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒)) | ||
Theorem | cbvrexdva 3407* | Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒)) | ||
Theorem | cbvral2vw 3408* | Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3411 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvrex2vw 3409* | Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3412 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvral3vw 3410* | Change bound variables of triple restricted universal quantification, using implicit substitution. Version of cbvral3v 3413 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 10-May-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) | ||
Theorem | cbvral2v 3411* | Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvral2vw 3408 when possible. (Contributed by NM, 10-Aug-2004.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvrex2v 3412* | Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrex2vw 3409 when possible. (Contributed by FL, 2-Jul-2012.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvral3v 3413* | Change bound variables of triple restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvral3vw 3410 when possible. (Contributed by NM, 10-May-2005.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) | ||
Theorem | cbvralsvw 3414* | Change bound variable by using a substitution. Version of cbvralsv 3416 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 20-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvrexsvw 3415* | Change bound variable by using a substitution. Version of cbvrexsv 3417 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 2-Mar-2008.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvralsv 3416* | Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvralsvw 3414 when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvrexsv 3417* | Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrexsvw 3415 when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | sbralie 3418* | Implicit to explicit substitution that swaps variables in a rectrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) | ||
Theorem | rabbiia 3419 | Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabbii 3420 | Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3427. (Contributed by Peter Mazsa, 1-Nov-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabbida 3421 | Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3425 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbid 3422 | Version of rabbidv 3427 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbidva2 3423* | Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabbia2 3424 | Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒} | ||
Theorem | rabbidva 3425* | Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.) (Proof shortened by SN, 3-Dec-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbidvaOLD 3426* | Obsolete proof of rabbidva 3425 as of 4-Dec-2023. (Contributed by NM, 28-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbidv 3427* | Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabeqf 3428 | Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | ||
Theorem | rabeqi 3429 | Equality theorem for restricted class abstractions. Inference form of rabeqf 3428. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2142, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 3-Jun-2024.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} | ||
Theorem | rabeqiOLD 3430 | Obsolete version of rabeqi 3429 as of 3-Jun-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2142 and ax-11 2158. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} | ||
Theorem | rabeq 3431* | Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2142, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | ||
Theorem | rabeqdv 3432* | Equality of restricted class abstractions. Deduction form of rabeq 3431. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) | ||
Theorem | rabeqbidv 3433* | Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabeqbidva 3434* | Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabeq2i 3435 | Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} ⇒ ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | rabswap 3436 | Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} | ||
Theorem | cbvrabw 3437* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3438 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | cbvrab 3438 | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrabw 3437 when possible. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | cbvrabv 3439* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2158 and ax-13 2379. (Revised by Steven Nguyen, 4-Dec-2022.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabrabi 3440* | Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2142 and ax-11 2158. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) ⇒ ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} | ||
Syntax | cvv 3441 | Extend class notation to include the universal class symbol. |
class V | ||
Theorem | vjust 3442 | Soundness justification theorem for df-v 3443. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} | ||
Definition | df-v 3443 | Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. The class V can be described as the "class of all sets"; vprc 5183 proves that V is not itself a set in ZFC. We will frequently use the expression 𝐴 ∈ V as a short way to say "𝐴 is a set", and isset 3453 proves that this expression has the same meaning as ∃𝑥𝑥 = 𝐴. The class V is called the "von Neumann universe", however, the letter "V" is not a tribute to the name of von Neumann. The letter "V" was used earlier by Peano in 1889 for the universe of sets, where the letter V is derived from the word "Verum". Peano's notation V was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. For a general discussion of the theory of classes, see mmset.html#class 3453. (Contributed by NM, 26-May-1993.) |
⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | ||
Theorem | vex 3444 | All setvar variables are sets (see isset 3453). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2175. (Revised by SN, 28-Aug-2023.) |
⊢ 𝑥 ∈ V | ||
Theorem | vexOLD 3445 | Obsolete version of vex 3444 as of 28-Aug-2023. All setvar variables are sets (see isset 3453). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑥 ∈ V | ||
Theorem | elv 3446 | If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3444), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
⊢ (𝑥 ∈ V → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | elvd 3447 | If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3444) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3446. (Contributed by Peter Mazsa, 23-Oct-2018.) |
⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | el2v 3448 | If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3444), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | eqv 3449* | The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2142, ax-11 2158, ax-13 2379. (Revised by BJ, 10-Aug-2022.) |
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | eqvf 3450 | The universe contains every set. (Contributed by BJ, 15-Jul-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | abv 3451 | The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 34347) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) |
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | ||
Theorem | elisset 3452* | An element of a class exists. (Contributed by NM, 1-May-1995.) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | isset 3453* |
Two ways to say "𝐴 is a set": A class 𝐴 is a
member of the
universal class V (see df-v 3443)
if and only if the class 𝐴
exists (i.e. there exists some set 𝑥 equal to class 𝐴).
Theorem 6.9 of [Quine] p. 43.
Notational convention: We will use the
notational device "𝐴 ∈ V " to mean "𝐴 is a
set" very
frequently, for example in uniex 7447. Note that a class 𝐴 which
is
not a set is called a proper class. In some theorems, such as
uniexg 7446, in order to shorten certain proofs we use
the more general
antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean
"𝐴 is a set."
Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2870 requires that the expression substituted for 𝐵 not contain 𝑥. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | ||
Theorem | issetf 3454 | A version of isset 3453 that does not require 𝑥 and 𝐴 to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | ||
Theorem | isseti 3455* | A way to say "𝐴 is a set" (inference form). (Contributed by NM, 24-Jun-1993.) Remove dependencies on axioms. (Revised by BJ, 13-Jul-2019.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | issetiOLD 3456* | Obsolete version of isseti 3455 as of 28-Aug-2023. A way to say "𝐴 is a set" (inference form). (Contributed by NM, 24-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | issetri 3457* | A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑥 𝑥 = 𝐴 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | eqvisset 3458 | A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3453 and issetri 3457. (Contributed by BJ, 27-Apr-2019.) |
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | ||
Theorem | elex 3459 | If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | ||
Theorem | elexi 3460 | If a class is a member of another class, then it is a set. Inference associated with elex 3459. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | elexd 3461 | If a class is a member of another class, then it is a set. Deduction associated with elex 3459. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | elissetOLD 3462* | Obsolete version of elisset 3452 as of 28-Aug-2023. An element of a class exists. (Contributed by NM, 1-May-1995.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | elex2 3463* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) | ||
Theorem | elex22 3464* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | ||
Theorem | prcnel 3465 | A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.) |
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) | ||
Theorem | ralv 3466 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | rexv 3467 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | reuv 3468 | A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | ||
Theorem | rmov 3469 | An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) | ||
Theorem | rabab 3470 | A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | ||
Theorem | rexcom4b 3471* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) | ||
Theorem | ralcom4OLD 3472* | Obsolete version of ralcom4 3198 as of 26-Aug-2023. Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexcom4OLD 3473* | Obsolete version of rexcom4 3212 as of 26-Aug-2023. Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | ceqsalt 3474* | Closed theorem version of ceqsalg 3476. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsralt 3475* | Restricted quantifier version of ceqsalt 3474. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsalg 3476* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3477. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsalgALT 3477* | Alternate proof of ceqsalg 3476, not using ceqsalt 3474. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsal 3478* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) | ||
Theorem | ceqsalv 3479* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) | ||
Theorem | ceqsralv 3480* | Restricted quantifier version of ceqsalv 3479. (Contributed by NM, 21-Jun-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | gencl 3481* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵)) & ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (𝜃 → 𝜓) | ||
Theorem | 2gencl 3482* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐶) & ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐷) & ⊢ (𝐴 = 𝐶 → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = 𝐷 → (𝜓 ↔ 𝜒)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝜑) ⇒ ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒) | ||
Theorem | 3gencl 3483* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐷) & ⊢ (𝐹 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐹) & ⊢ (𝐺 ∈ 𝑆 ↔ ∃𝑧 ∈ 𝑅 𝐶 = 𝐺) & ⊢ (𝐴 = 𝐷 → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = 𝐹 → (𝜓 ↔ 𝜒)) & ⊢ (𝐶 = 𝐺 → (𝜒 ↔ 𝜃)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅) → 𝜑) ⇒ ⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝜃) | ||
Theorem | cgsexg 3484* | Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.) |
⊢ (𝑥 = 𝐴 → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | cgsex2g 3485* | Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | cgsex4g 3486* | An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2142, ax-11 2158. (Revised by Gino Giotto, 28-Jun-2024.) |
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | cgsex4gOLD 3487* | Obsolete version of cgsex4g 3486 as of 28-Jun-2024. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2142. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒) & ⊢ (𝜒 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsex 3488* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | ceqsexv 3489* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | ceqsexv2d 3490* | Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ ∃𝑥𝜑 | ||
Theorem | ceqsex2 3491* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) | ||
Theorem | ceqsex2v 3492* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) Avoid ax-10 2142 and ax-11 2158. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) | ||
Theorem | ceqsex3v 3493* | Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃) | ||
Theorem | ceqsex4v 3494* | Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏) | ||
Theorem | ceqsex6v 3495* | Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐸 ∈ V & ⊢ 𝐹 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁) | ||
Theorem | ceqsex8v 3496* | Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐸 ∈ V & ⊢ 𝐹 ∈ V & ⊢ 𝐺 ∈ V & ⊢ 𝐻 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑡 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (𝑠 = 𝐻 → (𝜎 ↔ 𝜌)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌) | ||
Theorem | gencbvex 3497* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ 𝐴 ∈ V & ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) & ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) ⇒ ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓)) | ||
Theorem | gencbvex2 3498* | Restatement of gencbvex 3497 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.) |
⊢ 𝐴 ∈ V & ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) & ⊢ (𝜃 → ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) ⇒ ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓)) | ||
Theorem | gencbval 3499* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) |
⊢ 𝐴 ∈ V & ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) & ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) ⇒ ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓)) | ||
Theorem | sbhypf 3500* | Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3856. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
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