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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | raleqf 2501 | 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.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rexeqf 2502 | 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.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | reueq1f 2503 | Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rmoeq1f 2504 | Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | raleq 2505* | Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rexeq 2506* | Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) |
⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | reueq1 2507* | Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rmoeq1 2508* | Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | raleqi 2509* | Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑) | ||
Theorem | rexeqi 2510* | Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑) | ||
Theorem | raleqdv 2511* | Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rexeqdv 2512* | Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | raleqbi1dv 2513* | Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rexeqbi1dv 2514* | Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | reueqd 2515* | Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rmoeqd 2516* | Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | raleqbidv 2517* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbidv 2518* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | raleqbidva 2519* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | rexeqbidva 2520* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | mormo 2521 | Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | reu5 2522 | Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | reurex 2523 | Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | reurmo 2524 | Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo5 2525 | Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | nrexrmo 2526 | Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | cbvralf 2527 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrexf 2528 | Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvral 2529* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrex 2530* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreu 2531* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrmo 2532* | Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvralv 2533* | Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrexv 2534* | Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreuv 2535* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrmov 2536* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvraldva2 2537* | 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 2538* | 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.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) | ||
Theorem | cbvraldva 2539* | 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 2540* | 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 | cbvral2v 2541* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvrex2v 2542* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) | ||
Theorem | cbvral3v 2543* | Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.) |
⊢ (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 | cbvralsv 2544* | Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | cbvrexsv 2545* | Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | ||
Theorem | sbralie 2546* | Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) | ||
Theorem | rabbiia 2547 | Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabbidva 2548* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbidv 2549* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabeqf 2550 | 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 | rabeq 2551* | Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) |
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | ||
Theorem | rabeqbidv 2552* | 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 2553* | 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 2554 | Inference rule 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 | cbvrab 2555 | 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. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | cbvrabv 2556* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Syntax | cvv 2557 | Extend class notation to include the universal class symbol. |
class V | ||
Theorem | vjust 2558 | Soundness justification theorem for df-v 2559. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} | ||
Definition | df-v 2559 | Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.) |
⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | ||
Theorem | vex 2560 | All setvar variables are sets (see isset 2561). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝑥 ∈ V | ||
Theorem | isset 2561* |
Two ways to say "𝐴 is a set": A class 𝐴 is a
member of the
universal class V (see df-v 2559)
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 4174. Note the when 𝐴 is not
a set,
it is called a proper class. In some theorems, such as uniexg 4175, 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 2036 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 2562 | A version of isset 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 2563* | A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | issetri 2564* | A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
⊢ ∃𝑥 𝑥 = 𝐴 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | eqvisset 2565 | A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2561 and issetri 2564. (Contributed by BJ, 27-Apr-2019.) |
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | ||
Theorem | elex 2566 | If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | ||
Theorem | elexi 2567 | If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | elisset 2568* | An element of a class exists. (Contributed by NM, 1-May-1995.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | elex22 2569* | 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 | elex2 2570* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) | ||
Theorem | ralv 2571 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | rexv 2572 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | reuv 2573 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | ||
Theorem | rmov 2574 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) | ||
Theorem | rabab 2575 | 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 | ralcom4 2576* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexcom4 2577* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexcom4a 2578* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | rexcom4b 2579* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) | ||
Theorem | ceqsalt 2580* | Closed theorem version of ceqsalg 2582. (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 2581* | Restricted quantifier version of ceqsalt 2580. (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 2582* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | ceqsal 2583* | 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 2584* | 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 2585* | Restricted quantifier version of ceqsalv 2584. (Contributed by NM, 21-Jun-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | ||
Theorem | gencl 2586* | 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 2587* | 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 2588* | 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 2589* | 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 2590* | 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 2591* | An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) |
⊢ (((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 2592* | 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 2593* | 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 | ceqsex2 2594* | 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 2595* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) | ||
Theorem | ceqsex3v 2596* | 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 2597* | 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 2598* | 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 2599* | 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 2600* | 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)) |
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