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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremnfabdOLD 3001 Obsolete version of nfabd 2998 as of 10-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})
 
Theoremdvelimdc 3002 Deduction form of dvelimc 3003. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
 
Theoremdvelimc 3003 Version of dvelim 2465 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)
 
Theoremnfcvf 3004 If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-ext 2790. (Revised by Wolf Lammen, 10-May-2023.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
 
Theoremnfcvf2 3005 If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
 
TheoremnfcvfOLD 3006 Obsolete version of nfcvf 3004 as of 10-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
 
Theoremcleqf 3007 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2812. See also cleqh 2933. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2381. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2136. (Revised by Gino Giotto, 20-Aug-2023.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
TheoremcleqfOLD 3008 Obsolete version of cleqf 3007 as of 10-May-2023. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremabid2f 3009 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 3010 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 2381. (Revised by Wolf Lammen, 13-May-2023.)
𝑥𝐴       (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremabeq2fOLD 3011 Obsolete version of abeq2f 3010 as of 13-May-2023. (Contributed by Thierry Arnoux, 11-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴       (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremsbabel 3012* 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 3013 Extend wff notation to include inequality.
wff 𝐴𝐵
 
Definitiondf-ne 3014 Define inequality. (Contributed by NM, 26-May-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
 
Theoremneii 3015 Inference associated with df-ne 3014. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵
 
Theoremneir 3016 Inference associated with df-ne 3014. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵
 
Theoremnne 3017 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
𝐴𝐵𝐴 = 𝐵)
 
Theoremneneqd 3018 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)
 
Theoremneneq 3019 From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴𝐵 → ¬ 𝐴 = 𝐵)
 
Theoremneqned 3020 If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 3018. One-way deduction form of df-ne 3014. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 3039. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)
 
Theoremneqne 3021 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)
 
Theoremneirr 3022 No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)
¬ 𝐴𝐴
 
Theoremexmidne 3023 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
(𝐴 = 𝐵𝐴𝐵)
 
Theoremeqneqall 3024 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴 = 𝐵 → (𝐴𝐵𝜑))
 
Theoremnonconne 3025 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
¬ (𝐴 = 𝐵𝐴𝐵)
 
Theoremnecon3ad 3026 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 3027 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 3028 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 3029 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
 
Theoremnecon1ad 3030 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))
 
Theoremnecon1bd 3031 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 3032 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 3033 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 3034 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))
 
Theoremnecon1d 3035 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 3036 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(𝜑 → (𝐴 = 𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴𝐵))
 
Theoremnecon4d 3037 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 3038 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)       (𝐴𝐵 → ¬ 𝜑)
 
Theoremnecon3bi 3039 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 3040 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)
 
Theoremnecon1bi 3041 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 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.)
(𝐴 = 𝐵 → ¬ 𝜑)       (𝜑𝐴𝐵)
 
Theoremnecon2bi 3043 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 → ¬ 𝜑)
 
Theoremnecon4ai 3044 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 3045 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴𝐵)
 
Theoremnecon1i 3046 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴 = 𝐵)
 
Theoremnecon2i 3047 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴 = 𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴𝐵)
 
Theoremnecon4i 3048 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 3049 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
 
Theoremnecon3bbid 3050 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (¬ 𝜓𝐴𝐵))
 
Theoremnecon1abid 3051 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))
 
Theoremnecon1bbid 3052 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
(𝜑 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))
 
Theoremnecon4abid 3053 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 3054 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
(𝜑 → (¬ 𝜓𝐴𝐵))       (𝜑 → (𝜓𝐴 = 𝐵))
 
Theoremnecon2abid 3055 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 3056 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 3057 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 3058 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremnecon3abii 3059 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(𝐴 = 𝐵𝜑)       (𝐴𝐵 ↔ ¬ 𝜑)
 
Theoremnecon3bbii 3060 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴 = 𝐵)       𝜑𝐴𝐵)
 
Theoremnecon1abii 3061 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)
 
Theoremnecon1bbii 3062 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝐴𝐵𝜑)       𝜑𝐴 = 𝐵)
 
Theoremnecon2abii 3063 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
(𝐴 = 𝐵 ↔ ¬ 𝜑)       (𝜑𝐴𝐵)
 
Theoremnecon2bbii 3064 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 ↔ ¬ 𝜑)
 
Theoremnecon3bii 3065 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴𝐵𝐶𝐷)
 
Theoremnecom 3066 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(𝐴𝐵𝐵𝐴)
 
Theoremnecomi 3067 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
𝐴𝐵       𝐵𝐴
 
Theoremnecomd 3068 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)
 
Theoremnesym 3069 Characterization of inequality in terms of reversed equality (see bicom 223). (Contributed by BJ, 7-Jul-2018.)
(𝐴𝐵 ↔ ¬ 𝐵 = 𝐴)
 
Theoremnesymi 3070 Inference associated with nesym 3069. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵        ¬ 𝐵 = 𝐴
 
Theoremnesymir 3071 Inference associated with nesym 3069. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
¬ 𝐴 = 𝐵       𝐵𝐴
 
Theoremneeq1d 3072 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2d 3073 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq12d 3074 Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremneeq1 3075 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2 3076 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq1i 3077 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremneeq2i 3078 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremneeq12i 3079 Inference for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theoremeqnetrd 3080 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqnetrrd 3081 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)
 
Theoremneeqtrd 3082 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeqnetri 3083 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqnetrri 3084 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶
 
Theoremneeqtri 3085 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremneeqtrri 3086 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremneeqtrrd 3087 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theoremeqnetrrid 3088 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 3089 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 3090 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 3091 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3netr4g 3092 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theoremnebi 3093 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))
 
Theorempm13.18 3094 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)
 
Theorempm13.18OLD 3095 Obsolete version of pm13.18 3094 as of 14-May-2023. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
 
Theorempm13.181 3096 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorempm2.61ine 3097 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 3098 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)
 
Theorempm2.61ne 3099 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 3100 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)
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