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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eueq2 3701* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) | ||
Theorem | eueq3 3702* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) | ||
Theorem | moeq3 3703* | "At most one" property of equality (split into 3 cases). (The first two hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) | ||
Theorem | mosub 3704* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) | ||
Theorem | mo2icl 3705* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) | ||
Theorem | mob2 3706* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) | ||
Theorem | moi2 3707* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) | ||
Theorem | mob 3708* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)) | ||
Theorem | moi 3709* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵) | ||
Theorem | morex 3710* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) | ||
Theorem | euxfr2w 3711* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3713 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ 𝐴 ∈ V & ⊢ ∃*𝑦 𝑥 = 𝐴 ⇒ ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | ||
Theorem | euxfrw 3712* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3714 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | euxfr2 3713* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker euxfr2w 3711 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ ∃*𝑦 𝑥 = 𝐴 ⇒ ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | ||
Theorem | euxfr 3714* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker euxfrw 3712 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | euind 3715* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)) | ||
Theorem | reu2 3716* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | reu6 3717* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | reu3 3718* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))) | ||
Theorem | reu6i 3719* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | eqreu 3720* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo4 3721* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | reu4 3722* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | reu7 3723* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
Theorem | reu8 3724* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
Theorem | rmo3f 3725* | Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | rmo4f 3726* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | reu2eqd 3727* | Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | reueq 3728* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) | ||
Theorem | rmoeq 3729* | Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.) |
⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 | ||
Theorem | rmoan 3730 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) | ||
Theorem | rmoim 3731 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | rmoimia 3732 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmoimi 3733 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmoimi2 3734 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reu5a 3735 | Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑) ∧ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))) | ||
Theorem | reuimrmo 3736 | Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2700. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2reuswap 3737* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2reuswap2 3738* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | reuxfrd 3739* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) Separate variables B and C. (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓)) | ||
Theorem | reuxfr 3740* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) ⇒ ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑) | ||
Theorem | reuxfr1d 3741* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3742. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | reuxfr1ds 3742* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5320 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | reuxfr1 3743* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5321 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓) | ||
Theorem | reuind 3744* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ((∀𝑥∀𝑦(((𝐴 ∈ 𝐶 ∧ 𝜑) ∧ (𝐵 ∈ 𝐶 ∧ 𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴 ∈ 𝐶 ∧ 𝜑)) → ∃!𝑧 ∈ 𝐶 ∀𝑥((𝐴 ∈ 𝐶 ∧ 𝜑) → 𝑧 = 𝐴)) | ||
Theorem | 2rmorex 3745* | Double restricted quantification with "at most one", analogous to 2moex 2725. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reu5lem1 3746* | Lemma for 2reu5 3749. Note that ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵𝜑 does not mean "there is exactly one 𝑥 in 𝐴 and exactly one 𝑦 in 𝐵 such that 𝜑 holds"; see comment for 2eu5 2740. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | 2reu5lem2 3747* | Lemma for 2reu5 3749. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | 2reu5lem3 3748* | Lemma for 2reu5 3749. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3870. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2reu5 3749* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2740 and reu3 3718. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2reurex 3750* | Double restricted quantification with existential uniqueness, analogous to 2euex 2726. (Contributed by Alexander van der Vekens, 24-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reurmo 3751* | Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2727. (Contributed by Alexander van der Vekens, 24-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
Theorem | 2rmoswap 3752* | A condition allowing to swap restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2729. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2rexreu 3753* | Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2731. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
This is a very useless definition, which "abbreviates" (𝑥 = 𝑦 → 𝜑) as CondEq(𝑥 = 𝑦 → 𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation (𝑥 = 𝑦 → 𝜑). This is all used as part of a metatheorem: we want to say that ⊢ (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and ⊢ (𝑥 = 𝑦 → 𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables assumption: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦 → 𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜑)), which is provable by cdeqth 3758 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one setvar variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 3763 and cdeqab 3761. | ||
Syntax | wcdeq 3754 | Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result. |
wff CondEq(𝑥 = 𝑦 → 𝜑) | ||
Definition | df-cdeq 3755 | Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | ||
Theorem | cdeqi 3756 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqri 3757 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqth 3758 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝜑 ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqnot 3759 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) | ||
Theorem | cdeqal 3760* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
Theorem | cdeqab 3761* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) | ||
Theorem | cdeqal1 3762* | Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | cdeqab1 3763* | Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) | ||
Theorem | cdeqim 3764 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | cdeqcv 3765 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | cdeqeq 3766 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | cdeqel 3767 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | nfcdeq 3768* | If we have a conditional equality proof, where 𝜑 is 𝜑(𝑥) and 𝜓 is 𝜑(𝑦), and 𝜑(𝑥) in fact does not have 𝑥 free in it according to Ⅎ, then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that Ⅎ𝑥𝜑 is actually a not-free predicate. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜑 ↔ 𝜓) | ||
Theorem | nfccdeq 3769* | Variation of nfcdeq 3768 for classes. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2161. (Revised by Gino Giotto, 19-May-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | rru 3770* |
Relative version of Russell's paradox ru 3771 (which corresponds to the
case 𝐴 = V).
Originally a subproof in pwnss 5250. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3124. (Revised by Steven Nguyen, 23-Nov-2022.) |
⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 | ||
Theorem | ru 3771 |
Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system, which Frege acknowledged in the second edition of his Grundgesetze der Arithmetik. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 5225 asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. However, Zermelo was then faced with a "chicken and egg" problem of how to show 𝐵 is a set, leading him to introduce the set-building axioms of Null Set 0ex 5211, Pairing prex 5333, Union uniex 7467, Power Set pwex 5281, and Infinity omex 9106 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 6441 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types". Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ac4 9897 and Cantor's theorem canth 7111 are provably false. (See ncanth 7112 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 5203 replaces ax-rep 5190) with ax-sep 5203 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic", J. Symb. Logic 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 9060 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9066). See ruALT 9067 for an alternate proof of ru 3771 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2390. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Syntax | wsbc 3772 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class 𝐴 for setvar variable 𝑥 in wff 𝜑". |
wff [𝐴 / 𝑥]𝜑 | ||
Definition | df-sbc 3773 |
Define the proper substitution of a class for a set.
When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3782). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3802 for our definition, whose right-hand side always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3774 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 𝐴 rather than our falsehood. (This can be seen by substituting 𝐴, 𝑦, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic breakdown of 𝜑, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3774, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3773 in the form of sbc8g 3780. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of 𝐴 in every use of this definition) we allow direct reference to df-sbc 3773 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3781 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3774. The related definition df-csb 3884 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | dfsbcq 3774 |
Proper substitution of a class for a set in a wff given equal classes.
This is the essence of the sixth axiom of Frege, specifically Proposition
52 of [Frege1879] p. 50.
This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3773 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3775 instead of df-sbc 3773. (dfsbcq2 3775 is needed because unlike Quine we do not overload the df-sb 2070 syntax.) As a consequence of these theorems, we can derive sbc8g 3780, which is a weaker version of df-sbc 3773 that leaves substitution undefined when 𝐴 is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3780, so we will allow direct use of df-sbc 3773 after theorem sbc2or 3781 below. Proper substitution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | ||
Theorem | dfsbcq2 3775 | This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 2070 and substitution for class variables df-sbc 3773. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3774. (Contributed by NM, 31-Dec-2016.) |
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbsbc 3776 | Show that df-sb 2070 and df-sbc 3773 are equivalent when the class term 𝐴 in df-sbc 3773 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2070 for proofs involving df-sbc 3773. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbceq1d 3777 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | ||
Theorem | sbceq1dd 3778 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | ||
Theorem | sbceqbid 3779* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | ||
Theorem | sbc8g 3780 | This is the closest we can get to df-sbc 3773 if we start from dfsbcq 3774 (see its comments) and dfsbcq2 3775. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | sbc2or 3781* | The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [𝐴 / 𝑥]𝜑 behavior at proper classes, matching the sbc5 3800 (false) and sbc6 3802 (true) conclusions. This is interesting since dfsbcq 3774 and dfsbcq2 3775 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable 𝑦 that 𝜑 or 𝐴 may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.) |
⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | ||
Theorem | sbcex 3782 | By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | ||
Theorem | sbceq1a 3783 | Equality theorem for class substitution. Class version of sbequ12 2253. (Contributed by NM, 26-Sep-2003.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbceq2a 3784 | Equality theorem for class substitution. Class version of sbequ12r 2254. (Contributed by NM, 4-Jan-2017.) |
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | spsbc 3785 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2073 and rspsbc 3862. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | ||
Theorem | spsbcd 3786 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 2073 and rspsbc 3862. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcth 3787 | A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.) |
⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcthdv 3788* | Deduction version of sbcth 3787. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcid 3789 | An identity theorem for substitution. See sbid 2257. (Contributed by Mario Carneiro, 18-Feb-2017.) |
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | nfsbc1d 3790 | Deduction version of nfsbc1 3791. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) | ||
Theorem | nfsbc1 3791 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbc1v 3792* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbcdw 3793* | Deduction version of nfsbcw 3794. Version of nfsbcd 3796 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | ||
Theorem | nfsbcw 3794* | Bound-variable hypothesis builder for class substitution. Version of nfsbc 3797 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 7-Sep-2014.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 | ||
Theorem | sbccow 3795* | A composition law for class substitution. Version of sbcco 3798 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | nfsbcd 3796 | Deduction version of nfsbc 3797. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfsbcdw 3793 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | ||
Theorem | nfsbc 3797 | Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfsbcw 3794 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 | ||
Theorem | sbcco 3798* | A composition law for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker sbccow 3795 when possible. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.) |
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcco2 3799* | A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbc5 3800* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
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