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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nfcxfrd 2901 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfcv 2902* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcvd 2903* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfab1 2904 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | ||
Theorem | nfnfc1 2905 | The setvar 𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | ||
Theorem | clelsb1fw 2906* | Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2113). Version of clelsb1f 2907 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Rodolfo Medina, 28-Apr-2010.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | ||
Theorem | clelsb1f 2907 | Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2113). Usage of this theorem is discouraged because it depends on ax-13 2374. See clelsb1fw 2906 not requiring ax-13 2374, but extra disjoint variables. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof shortened by Wolf Lammen, 7-May-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | ||
Theorem | nfab 2908* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2374. See nfabg 2909 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑} | ||
Theorem | nfabg 2909 | Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfab 2908 for a version with more disjoint variable conditions, but not requiring ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑} | ||
Theorem | nfaba1 2910* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2374. See nfaba1g 2912 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1964, ax-7 2004, ax-12 2174. (Revised by SN, 14-May-2025.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfaba1OLD 2911* | Obsolete version of nfaba1 2910 as of 14-May-2025. (Contributed by Mario Carneiro, 14-Oct-2016.) (Revised by GG, 20-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfaba1g 2912 | Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfaba1 2910 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.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfeqd 2913 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) | ||
Theorem | nfeld 2914 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) | ||
Theorem | nfnfc 2915 | Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2374. (Revised by Wolf Lammen, 10-Dec-2019.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 | ||
Theorem | nfeq 2916 | Hypothesis builder for equality. (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel 2917 | Hypothesis builder for elementhood. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq1 2918* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel1 2919* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq2 2920* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel2 2921* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | drnfc1 2922 | Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2107, ax-11 2154. (Revised by Wolf Lammen, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) | ||
Theorem | drnfc2 2923 | Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1917 with dral2 2440, leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 2440 depends on ax-13 2374, hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2107. (Revised by Wolf Lammen, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) | ||
Theorem | nfabdw 2924* | Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2925 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | nfabd 2925 | Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker nfabdw 2924 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2115 and ax-ext 2705. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | nfabd2 2926 | Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | dvelimdc 2927 | Deduction form of dvelimc 2928. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑧𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑧𝐵) & ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) | ||
Theorem | dvelimc 2928 | Version of dvelim 2453 for classes. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑧𝐵 & ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) | ||
Theorem | nfcvf 2929 | If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfcv 2902 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-ext 2705. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | ||
Theorem | nfcvf2 2930 | If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfcv 2902 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | ||
Theorem | cleqf 2931 | Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2727. See also cleqh 2868. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2374. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2138. (Revised by GG, 20-Aug-2023.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | eqabf 2932 | Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a nonfree variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2374. (Revised by Wolf Lammen, 13-May-2023.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | ||
Theorem | abid2f 2933 | A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Feb-2025.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | abid2fOLD 2934 | Obsolete version of abid2f 2933 as of 26-Feb-2025. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | sbabel 2935* | Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 28-Oct-2024.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴) | ||
Theorem | sbabelOLD 2936* | Obsolete version of sbabel 2935 as of 28-Oct-2024. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴) | ||
Syntax | wne 2937 | Extend wff notation to include inequality. |
wff 𝐴 ≠ 𝐵 | ||
Definition | df-ne 2938 | Define inequality. (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | ||
Theorem | neii 2939 | Inference associated with df-ne 2938. (Contributed by BJ, 7-Jul-2018.) |
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ ¬ 𝐴 = 𝐵 | ||
Theorem | neir 2940 | Inference associated with df-ne 2938. (Contributed by BJ, 7-Jul-2018.) |
⊢ ¬ 𝐴 = 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | nne 2941 | Negation of inequality. (Contributed by NM, 9-Jun-2006.) |
⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | neneqd 2942 | Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
Theorem | neneq 2943 | From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | ||
Theorem | neqned 2944 | If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 2942. One-way deduction form of df-ne 2938. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2964. (Revised by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | neqne 2945 | From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | ||
Theorem | neirr 2946 | No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ ¬ 𝐴 ≠ 𝐴 | ||
Theorem | exmidne 2947 | Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) | ||
Theorem | eqneqall 2948 | A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) | ||
Theorem | nonconne 2949 | Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.) |
⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) | ||
Theorem | necon3ad 2950 | Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓)) | ||
Theorem | necon3bd 2951 | Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon2ad 2952 | Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon2bd 2953 | Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.) |
⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) | ||
Theorem | necon1ad 2954 | Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (¬ 𝜓 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓)) | ||
Theorem | necon1bd 2955 | Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → 𝐴 = 𝐵)) | ||
Theorem | necon4ad 2956 | Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵)) | ||
Theorem | necon4bd 2957 | Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) | ||
Theorem | necon3d 2958 | Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.) |
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon1d 2959 | Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)) | ||
Theorem | necon2d 2960 | Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.) |
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon4d 2961 | Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵)) | ||
Theorem | necon3ai 2962 | Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 28-Oct-2024.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) | ||
Theorem | necon3aiOLD 2963 | Obsolete version of necon3ai 2962 as of 28-Oct-2024. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) | ||
Theorem | necon3bi 2964 | Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 = 𝐵 → 𝜑) ⇒ ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | necon1ai 2965 | Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (¬ 𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝐴 ≠ 𝐵 → 𝜑) | ||
Theorem | necon1bi 2966 | Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 → 𝜑) ⇒ ⊢ (¬ 𝜑 → 𝐴 = 𝐵) | ||
Theorem | necon2ai 2967 | Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 = 𝐵 → ¬ 𝜑) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | necon2bi 2968 | Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 → ¬ 𝜑) | ||
Theorem | necon4ai 2969 | Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | necon3i 2970 | Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵) | ||
Theorem | necon1i 2971 | Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) |
⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ (𝐶 ≠ 𝐷 → 𝐴 = 𝐵) | ||
Theorem | necon2i 2972 | Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) |
⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵) | ||
Theorem | necon4i 2973 | Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵) | ||
Theorem | necon3abid 2974 | Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.) |
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)) | ||
Theorem | necon3bbid 2975 | Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.) |
⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | necon1abid 2976 | Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) | ||
Theorem | necon1bbid 2977 | Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) | ||
Theorem | necon4abid 2978 | Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓)) | ||
Theorem | necon4bbid 2979 | Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.) |
⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵)) | ||
Theorem | necon2abid 2980 | Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | necon2bbid 2981 | Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓)) | ||
Theorem | necon3bid 2982 | Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) | ||
Theorem | necon4bid 2983 | Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) | ||
Theorem | necon3abii 2984 | Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.) |
⊢ (𝐴 = 𝐵 ↔ 𝜑) ⇒ ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑) | ||
Theorem | necon3bbii 2985 | Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.) |
⊢ (𝜑 ↔ 𝐴 = 𝐵) ⇒ ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵) | ||
Theorem | necon1abii 2986 | Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵) ⇒ ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑) | ||
Theorem | necon1bbii 2987 | Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 ↔ 𝜑) ⇒ ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵) | ||
Theorem | necon2abii 2988 | Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.) |
⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑) ⇒ ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵) | ||
Theorem | necon2bbii 2989 | Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.) |
⊢ (𝜑 ↔ 𝐴 ≠ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑) | ||
Theorem | necon3bii 2990 | Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.) |
⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ⇒ ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) | ||
Theorem | necom 2991 | Commutation of inequality. (Contributed by NM, 14-May-1999.) |
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | ||
Theorem | necomi 2992 | Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 | ||
Theorem | necomd 2993 | Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) | ||
Theorem | nesym 2994 | Characterization of inequality in terms of reversed equality (see bicom 222). (Contributed by BJ, 7-Jul-2018.) |
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) | ||
Theorem | nesymi 2995 | Inference associated with nesym 2994. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ ¬ 𝐵 = 𝐴 | ||
Theorem | nesymir 2996 | Inference associated with nesym 2994. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
⊢ ¬ 𝐴 = 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 | ||
Theorem | neeq1d 2997 | Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | ||
Theorem | neeq2d 2998 | Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) | ||
Theorem | neeq12d 2999 | Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) | ||
Theorem | neeq1 3000 | Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
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