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Type | Label | Description | ||||||||||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||||||||||

Syntax | ccosh 28201 | Extend class notation to include the hyperbolic cosine function. see df-cosh 28204. | ||||||||||||||||||||||||||||||

cosh | ||||||||||||||||||||||||||||||||

Syntax | ctanh 28202 | Extend class notation to include the hyperbolic tangent function, see df-tanh 28205. | ||||||||||||||||||||||||||||||

tanh | ||||||||||||||||||||||||||||||||

Definition | df-sinh 28203 | Define the hyperbolic sine function (sinh). We define it this way for cmpt 4077, which requires the form . See sinhval-named 28206 for a simple way to evaluate it. We define this function by dividing by , which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 28209 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) | ||||||||||||||||||||||||||||||

sinh | ||||||||||||||||||||||||||||||||

Definition | df-cosh 28204 | Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4077, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.) | ||||||||||||||||||||||||||||||

cosh | ||||||||||||||||||||||||||||||||

Definition | df-tanh 28205 | Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4077, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.) | ||||||||||||||||||||||||||||||

tanh cosh | ||||||||||||||||||||||||||||||||

Theorem | sinhval-named 28206 | Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 28203. See sinhval 12434 for a theorem to convert this further. See sinh-conventional 28209 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) | ||||||||||||||||||||||||||||||

sinh | ||||||||||||||||||||||||||||||||

Theorem | coshval-named 28207 | Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 28204. See coshval 12435 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.) | ||||||||||||||||||||||||||||||

cosh | ||||||||||||||||||||||||||||||||

Theorem | tanhval-named 28208 | Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 28205. (Contributed by David A. Wheeler, 10-May-2015.) | ||||||||||||||||||||||||||||||

cosh tanh | ||||||||||||||||||||||||||||||||

Theorem | sinh-conventional 28209 | Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.) | ||||||||||||||||||||||||||||||

sinh | ||||||||||||||||||||||||||||||||

Theorem | sinhpcosh 28210 | Prove that sinh cosh using the conventional hyperbolic trig functions. (Contributed by David A. Wheeler, 27-May-2015.) | ||||||||||||||||||||||||||||||

sinh cosh | ||||||||||||||||||||||||||||||||

18.24.4 Reciprocal trig functions (sec, csc,
cot)Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them. | ||||||||||||||||||||||||||||||||

Syntax | csec 28211 | Extend class notation to include the secant function, see df-sec 28214. | ||||||||||||||||||||||||||||||

Syntax | ccsc 28212 | Extend class notation to include the cosecant function, see df-csc 28215. | ||||||||||||||||||||||||||||||

Syntax | ccot 28213 | Extend class notation to include the cotangent function, see df-cot 28216. | ||||||||||||||||||||||||||||||

Definition | df-sec 28214* | Define the secant function. We define it this way for cmpt 4077, which requires the form . The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Definition | df-csc 28215* | Define the cosecant function. We define it this way for cmpt 4077, which requires the form . The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Definition | df-cot 28216* | Define the cotangent function. We define it this way for cmpt 4077, which requires the form . The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | secval 28217 | Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | cscval 28218 | Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | cotval 28219 | Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | seccl 28220 | The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | csccl 28221 | The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | cotcl 28222 | The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | reseccl 28223 | The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | recsccl 28224 | The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | recotcl 28225 | The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | recsec 28226 | The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | reccsc 28227 | The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | reccot 28228 | The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | rectan 28229 | The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | sec0 28230 | The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.) | ||||||||||||||||||||||||||||||

Theorem | onetansqsecsq 28231 | Prove the tangent squared secant squared identity A ) ^ 2 ) ) = ( ( sec . (Contributed by David A. Wheeler, 25-May-2015.) | ||||||||||||||||||||||||||||||

Theorem | cotsqcscsq 28232 | Prove the tangent squared cosecant squared identity A ) ^ 2 ) ) = ( ( csc . (Contributed by David A. Wheeler, 27-May-2015.) | ||||||||||||||||||||||||||||||

18.24.5 Identities for "if"Utility theorems for "if". | ||||||||||||||||||||||||||||||||

Theorem | ifnmfalse 28233 | If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3572 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

18.24.6 Not-member-of | ||||||||||||||||||||||||||||||||

Theorem | AnelBC 28234 | If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using . (Contributed by David A. Wheeler, 10-May-2015.) | ||||||||||||||||||||||||||||||

18.24.7 Decimal pointDefine the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 28238 and df-dp2 28237 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10125. TODO: Fix non-existent label dfpval. | ||||||||||||||||||||||||||||||||

Syntax | cdp2 28235 | Constant used for decimal fraction constructor. See df-dp2 28237. | ||||||||||||||||||||||||||||||

_ | ||||||||||||||||||||||||||||||||

Syntax | cdp 28236 | Decimal point operator. See df-dp 28238. | ||||||||||||||||||||||||||||||

Definition | df-dp2 28237 | Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10125. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

_ | ||||||||||||||||||||||||||||||||

Definition | df-dp 28238* |
Define the (decimal point) operator. For example,
, and
;__ ;;;; ;;;
Unary minus, if applied, should normally be applied in front of the
parentheses.
Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers. The RHS is , not ; this should simplify some proofs. The LHS is , since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

_ | ||||||||||||||||||||||||||||||||

Theorem | dp2cl 28239 | Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

_ | ||||||||||||||||||||||||||||||||

Theorem | dpval 28240 | Define the value of the decimal point operator. See df-dp 28238. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

_ | ||||||||||||||||||||||||||||||||

Theorem | dpcl 28241 | Prove that the closure of the decimal point is as we have defined it. See df-dp 28238. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

Theorem | dpfrac1 28242 | Prove a simple equivalence involving the decimal point. See df-dp 28238 and dpcl 28241. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

; | ||||||||||||||||||||||||||||||||

18.24.8 Signum (sgn or sign)
function | ||||||||||||||||||||||||||||||||

Syntax | csgn 28243 | Extend class notation to include the Signum function. | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Definition | df-sgn 28244 | Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over (df-xr 8871) instead of so that it can accept and . Note that df-psgn 27415 defines the sign of a permutation, which is different. Value shown in sgnval 28245. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgnval 28245 | Value of Signum function. Pronounced "signum" . See df-sgn 28244. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgn0 28246 | Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgnp 28247 | Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgnrrp 28248 | Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgn1 28249 | Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgnpnf 28250 | Proof that the signum of is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgnn 28251 | Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

Theorem | sgnmnf 28252 | Proof that the signum of is -1. (Contributed by David A. Wheeler, 26-Jun-2016.) | ||||||||||||||||||||||||||||||

sgn | ||||||||||||||||||||||||||||||||

18.24.9 Ceiling function | ||||||||||||||||||||||||||||||||

Syntax | ccei 28253 | Extend class notation to include the ceiling function. | ||||||||||||||||||||||||||||||

⌈ | ||||||||||||||||||||||||||||||||

Definition | df-ceiling 28254 |
The ceiling function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and
the "NIST Digital Library of Mathematical Functions" , front
introduction,
"Common Notations and Definitions" section at
http://dlmf.nist.gov/front/introduction#Sx4.
By convention metamath users tend to expand this construct directly, instead of using the definition. However, we want to make sure this is separately and formally defined. Proof ceicl 10955 shows that the ceiling function returns an integer when provided a real. Formalized by David A. Wheeler. (Contributed by David A. Wheeler, 19-May-2015.) | ||||||||||||||||||||||||||||||

⌈ | ||||||||||||||||||||||||||||||||

Theorem | ceilingval 28255 | The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) | ||||||||||||||||||||||||||||||

⌈ | ||||||||||||||||||||||||||||||||

Theorem | ceilingcl 28256 | Closure of the ceiling function; the real work is in ceicl 10955. (Contributed by David A. Wheeler, 19-May-2015.) | ||||||||||||||||||||||||||||||

⌈ | ||||||||||||||||||||||||||||||||

18.24.10 Logarithm laws generalized to an
arbitrary base - log_Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear. This supports the notational form logb; that looks a little more like traditional notation, but is different than other 2-parameter functions. E.G., log_;; | ||||||||||||||||||||||||||||||||

Syntax | clog_ 28257 | Extend class notation to include the logarithm generalized to an arbitrary base. | ||||||||||||||||||||||||||||||

log_ | ||||||||||||||||||||||||||||||||

Definition | df-log_ 28258* | Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as log_ for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.) | ||||||||||||||||||||||||||||||

log_ | ||||||||||||||||||||||||||||||||

18.24.11 MiscellaneousMiscellaneous proofs. | ||||||||||||||||||||||||||||||||

Theorem | 5m4e1 28259 | Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.) | ||||||||||||||||||||||||||||||

Theorem | 2p2ne5 28260 | Prove that . In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.) | ||||||||||||||||||||||||||||||

Theorem | resolution 28261 | Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.) | ||||||||||||||||||||||||||||||

18.25 Mathbox for Alan Sare | ||||||||||||||||||||||||||||||||

18.25.1 Conventional Metamath proofs, some
derived from VD proofs | ||||||||||||||||||||||||||||||||

Theorem | iidn3 28262 | idn3 28387 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee222 28263 | e222 28408 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee3bir 28264 | Right-biconditional form of e3 28512 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee13 28265 | e13 28523 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee121 28266 | e121 28428 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee122 28267 | e122 28425 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee333 28268 | e333 28508 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee323 28269 | e323 28541 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3ornot23 28270 | If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 28623. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | orbi1r 28271 | orbi1 686 with order of disjuncts reversed. Derived from orbi1rVD 28624. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | bitr3 28272 | Closed nested implication form of bitr3i 242. Derived automatically from bitr3VD 28625. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3orbi123 28273 | pm4.39 841 with a 3-conjunct antecedent. This proof is 3orbi123VD 28626 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | syl5imp 28274 | Closed form of syl5 28. Derived automatically from syl5impVD 28639. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | impexp3a 28275 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the
User's Proof was completed it was minimized. The completed User's Proof
before minimization is not shown. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | com3rgbi 28276 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | impexp3acom3r 28277 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||

Theorem | ee1111 28278 |
Non-virtual deduction form of e1111 28447. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||

Theorem | pm2.43bgbi 28279 |
Logical equivalence of a 2-left-nested implication and a 1-left-nested
implicated
when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||

Theorem | pm2.43cbi 28280 |
Logical equivalence of a 3-left-nested implication and a 2-left-nested
implicated when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is
a Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's
Metamath Proof Assistant. The completed Virtual Deduction Proof (not
shown) was minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||

Theorem | ee233 28281 |
Non-virtual deduction form of e233 28540. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||

Theorem | imbi12 28282 | Implication form of imbi12i 316. imbi12 28282 is imbi12VD 28649 without virtual deductions and was automatically derived from imbi12VD 28649 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | imbi13 28283 | Join three logical equivalences to form equivalence of implications. imbi13 28283 is imbi13VD 28650 without virtual deductions and was automatically derived from imbi13VD 28650 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ee33 28284 |
Non-virtual deduction form of e33 28509. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
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Theorem | con5 28285 | Bi-conditional contraposition variation. This proof is con5VD 28676 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | con5i 28286 | Inference form of con5 28285. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | exlimexi 28287 | Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sb5ALT 28288* | Equivalence for substitution. Alternate proof of sb5 2039. This proof is sb5ALTVD 28689 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | eexinst01 28289 | exinst01 28397 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | eexinst11 28290 | exinst11 28398 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | vk15.4j 28291 | Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 28291 is vk15.4jVD 28690 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | notnot2ALT 28292 | Converse of double negation. Alternate proof of notnot2 104. This proof is notnot2ALTVD 28691 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | con3ALT 28293 | Contraposition. Alternate proof of con3 126. This proof is con3ALTVD 28692 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | ssralv2 28294* |
Quantification restricted to a subclass for two quantifiers. ssralv 3237
for two quantifiers. The proof of ssralv2 28294 was automatically generated
by minimizing the automatically translated proof of ssralv2VD 28642. The
automatic translation is by the tools program
translate_{without}_overwriting.cmd
(Contributed by Alan Sare,
18-Feb-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | sbc3org 28295 | sbcorg 3036 with a 3-disjuncts. This proof is sbc3orgVD 28627 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | alrim3con13v 28296* | Closed form of alrimi 1745 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 28628 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | rspsbc2 28297* | rspsbc 3069 with two quantifying variables. This proof is rspsbc2VD 28631 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | sbcoreleleq 28298* | Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 28635. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | tratrb 28299* | If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 28637. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

Theorem | 3ax5 28300 | ax-5 1544 for a 3 element left-nested implication. Derived automatically from 3ax5VD 28638. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||

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