Home Metamath Proof ExplorerTheorem List (p. 31 of 453) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28699) Hilbert Space Explorer (28700-30222) Users' Mathboxes (30223-45272)

Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfabd 3001 Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker nfabdw 3000 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2124 and ax-ext 2794. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremnfabd2 3002 Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremdvelimdc 3003 Deduction form of dvelimc 3004. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Theoremdvelimc 3004 Version of dvelim 2474 for classes. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Theoremnfcvf 3005 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 2391. See nfcv 2979 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 2794. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Theoremnfcvf2 3006 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 2391. See nfcv 2979 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 = 𝑦𝑦𝑥)

Theoremcleqf 3007 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2816. See also cleqh 2937. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2391. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2145. (Revised by Gino Giotto, 20-Aug-2023.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremabid2f 3008 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, 17-Nov-2019.)
𝑥𝐴       {𝑥𝑥𝐴} = 𝐴

Theoremabeq2f 3009 Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2391. (Revised by Wolf Lammen, 13-May-2023.)
𝑥𝐴       (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremsbabel 3010* 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, 26-Dec-2019.)
𝑥𝐴       ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

2.1.4  Negated equality and membership

2.1.4.1  Negated equality

Syntaxwne 3011 Extend wff notation to include inequality.
wff 𝐴𝐵

Definitiondf-ne 3012 Define inequality. (Contributed by NM, 26-May-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)

Theoremneii 3013 Inference associated with df-ne 3012. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵

Theoremneir 3014 Inference associated with df-ne 3012. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵

Theoremnne 3015 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
𝐴𝐵𝐴 = 𝐵)

Theoremneneqd 3016 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)

Theoremneneq 3017 From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴𝐵 → ¬ 𝐴 = 𝐵)

Theoremneqned 3018 If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 3016. One-way deduction form of df-ne 3012. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 3037. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremneqne 3019 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)

Theoremneirr 3020 No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)
¬ 𝐴𝐴

Theoremexmidne 3021 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
(𝐴 = 𝐵𝐴𝐵)

Theoremeqneqall 3022 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴 = 𝐵 → (𝐴𝐵𝜑))

Theoremnonconne 3023 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
¬ (𝐴 = 𝐵𝐴𝐵)

Theoremnecon3ad 3024 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 → (𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵 → ¬ 𝜓))

Theoremnecon3bd 3025 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴𝐵))

Theoremnecon2ad 3026 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 → (𝜓𝐴𝐵))

Theoremnecon2bd 3027 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Theoremnecon1ad 3028 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))

Theoremnecon1bd 3029 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 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))

Theoremnecon4ad 3030 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 → (𝐴𝐵 → ¬ 𝜓))       (𝜑 → (𝜓𝐴 = 𝐵))

Theoremnecon4bd 3031 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 → (¬ 𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵𝜓))

Theoremnecon3d 3032 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))

Theoremnecon1d 3033 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴 = 𝐵))

Theoremnecon2d 3034 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(𝜑 → (𝐴 = 𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴𝐵))

Theoremnecon4d 3035 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴 = 𝐵))

Theoremnecon3ai 3036 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)       (𝐴𝐵 → ¬ 𝜑)

Theoremnecon3bi 3037 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 = 𝐵𝜑)       𝜑𝐴𝐵)

Theoremnecon1ai 3038 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)

Theoremnecon1bi 3039 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.)
(𝐴𝐵𝜑)       𝜑𝐴 = 𝐵)

Theoremnecon2ai 3040 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)       (𝜑𝐴𝐵)

Theoremnecon2bi 3041 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 → ¬ 𝜑)

Theoremnecon4ai 3042 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.)
(𝐴𝐵 → ¬ 𝜑)       (𝜑𝐴 = 𝐵)

Theoremnecon3i 3043 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴𝐵)

Theoremnecon1i 3044 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴 = 𝐵)

Theoremnecon2i 3045 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴 = 𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴𝐵)

Theoremnecon4i 3046 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.)
(𝐴𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴 = 𝐵)

Theoremnecon3abid 3047 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))

Theoremnecon3bbid 3048 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (¬ 𝜓𝐴𝐵))

Theoremnecon1abid 3049 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))

Theoremnecon1bbid 3050 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
(𝜑 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))

Theoremnecon4abid 3051 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))       (𝜑 → (𝐴 = 𝐵𝜓))

Theoremnecon4bbid 3052 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
(𝜑 → (¬ 𝜓𝐴𝐵))       (𝜑 → (𝜓𝐴 = 𝐵))

Theoremnecon2abid 3053 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))       (𝜑 → (𝜓𝐴𝐵))

Theoremnecon2bbid 3054 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Theoremnecon3bid 3055 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐴𝐵𝐶𝐷))

Theoremnecon4bid 3056 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Theoremnecon3abii 3057 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(𝐴 = 𝐵𝜑)       (𝐴𝐵 ↔ ¬ 𝜑)

Theoremnecon3bbii 3058 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴 = 𝐵)       𝜑𝐴𝐵)

Theoremnecon1abii 3059 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)

Theoremnecon1bbii 3060 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝐴𝐵𝜑)       𝜑𝐴 = 𝐵)

Theoremnecon2abii 3061 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
(𝐴 = 𝐵 ↔ ¬ 𝜑)       (𝜑𝐴𝐵)

Theoremnecon2bbii 3062 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 ↔ ¬ 𝜑)

Theoremnecon3bii 3063 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴𝐵𝐶𝐷)

Theoremnecom 3064 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(𝐴𝐵𝐵𝐴)

Theoremnecomi 3065 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
𝐴𝐵       𝐵𝐴

Theoremnecomd 3066 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)

Theoremnesym 3067 Characterization of inequality in terms of reversed equality (see bicom 225). (Contributed by BJ, 7-Jul-2018.)
(𝐴𝐵 ↔ ¬ 𝐵 = 𝐴)

Theoremnesymi 3068 Inference associated with nesym 3067. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵        ¬ 𝐵 = 𝐴

Theoremnesymir 3069 Inference associated with nesym 3067. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
¬ 𝐴 = 𝐵       𝐵𝐴

Theoremneeq1d 3070 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremneeq2d 3071 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremneeq12d 3072 Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremneeq1 3073 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneeq2 3074 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremneeq1i 3075 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremneeq2i 3076 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremneeq12i 3077 Inference for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theoremeqnetrd 3078 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremeqnetrrd 3079 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)

Theoremneeqtrd 3080 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremeqnetri 3081 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqnetrri 3082 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶

Theoremneeqtri 3083 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremneeqtrri 3084 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremneeqtrrd 3085 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theoremeqnetrrid 3086 A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theorem3netr3d 3087 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3netr4d 3088 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 21-Nov-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theorem3netr3g 3089 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3netr4g 3090 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theoremnebi 3091 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))

Theorempm13.18 3092 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 14-May-2023.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)

Theorempm13.181 3093 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)

Theorempm2.61ine 3094 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝐴 = 𝐵𝜑)    &   (𝐴𝐵𝜑)       𝜑

Theorempm2.21ddne 3095 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)

Theorempm2.61ne 3096 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝜓𝜒))    &   ((𝜑𝐴𝐵) → 𝜓)    &   (𝜑𝜒)       (𝜑𝜓)

Theorempm2.61dne 3097 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝜓))    &   (𝜑 → (𝐴𝐵𝜓))       (𝜑𝜓)

Theorempm2.61dane 3098 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐴𝐵) → 𝜓)       (𝜑𝜓)

Theorempm2.61da2ne 3099 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐶 = 𝐷) → 𝜓)    &   ((𝜑 ∧ (𝐴𝐵𝐶𝐷)) → 𝜓)       (𝜑𝜓)

Theorempm2.61da3ne 3100 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐶 = 𝐷) → 𝜓)    &   ((𝜑𝐸 = 𝐹) → 𝜓)    &   ((𝜑 ∧ (𝐴𝐵𝐶𝐷𝐸𝐹)) → 𝜓)       (𝜑𝜓)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45272
 Copyright terms: Public domain < Previous  Next >