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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcscval 27901 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 27902 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 27903 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 27904 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 27905 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 27906 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 27907 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 27908 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 27909 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 27910 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 27911 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 27912 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 27913 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 27914 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 27915 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
18.23.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 27916 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 3648 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
18.23.6  Not-member-of
 
TheoremAnelBC 27917 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |-  A  e/  { B ,  C }
 
18.23.7  Decimal point

Define 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 27921 and df-dp2 27920 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10214.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 27918 Constant used for decimal fraction constructor. See df-dp2 27920.
 class _ A B
 
Syntaxcdp 27919 Decimal point operator. See df-dp 27921.
 class  period
 
Definitiondf-dp2 27920 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10214. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 27921* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) 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  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, 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  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 27922 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 27923 Define the value of the decimal point operator. See df-dp 27921. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 27924 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 27921. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 27925 Prove a simple equivalence involving the decimal point. See df-dp 27921 and dpcl 27924. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
18.23.8  Signum (sgn or sign) function
 
Syntaxcsgn 27926 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 27927 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  RR* (df-xr 8958) instead of  RR so that it can accept  +oo and  -oo. Note that df-psgn 26738 defines the sign of a permutation, which is different. Value shown in sgnval 27928. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 , 
 0 ,  if ( x  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 27928 Value of Signum function. Pronounced "signum" . See df-sgn 27927. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 27929 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
 
Theoremsgnp 27930 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  0  <  A ) 
 ->  (sgn `  A )  =  1 )
 
Theoremsgnrrp 27931 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
 |-  ( A  e.  RR+  ->  (sgn `  A )  =  1 )
 
Theoremsgn1 27932 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn `  1 )  =  1
 
Theoremsgnpnf 27933 Proof that the signum of  +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 +oo )  =  1
 
Theoremsgnn 27934 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  A  <  0 ) 
 ->  (sgn `  A )  =  -u 1 )
 
Theoremsgnmnf 27935 Proof that the signum of  -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 -oo )  =  -u 1
 
18.23.9  Ceiling function
 
Syntaxccei 27936 Extend class notation to include the ceiling function.
 class
 
Definitiondf-ceiling 27937 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 11044 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.)

 |- =  ( x  e.  RR  |->  -u ( |_ `  -u x ) )
 
Theoremceilingval 27938 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceilingcl 27939 Closure of the ceiling function; the real work is in ceicl 11044. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  e.  ZZ )
 
18.23.10  Logarithms generalized to arbitrary base using ` logb `
 
Theoremene0 27940  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremene1 27941  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremelogb 27942 Using  _e as the base is the same as  log. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  ( A  e.  ( CC  \  { 0 } )  ->  ( _elogb A )  =  ( log `  A ) )
 
18.23.11  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  ( (log_ `  B ) `  X
); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.G.,  ( (log_ `  10 ) ` ;; 1 0 0 )  =  2

This form is less convenient to work with inside metamath as compared to the  ( Blogb X
) form defined separately.

 
Syntaxclog_ 27943 Extend class notation to include the logarithm generalized to an arbitrary base.
 class log_
 
Definitiondf-log_ 27944* Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as  ( (log_ `  B ) `  X ) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.)
 |- log_  =  ( b  e.  ( CC  \  { 0 ,  1 } )  |->  ( x  e.  ( CC  \  { 0 } )  |->  ( ( log `  x )  /  ( log `  b
 ) ) ) )
 
18.23.12  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 27945 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 27946 Prove that  2  +  2  =/=  5. 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  2  +  2  =  5 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.)
 |-  (
 2  +  2 )  =/=  5
 
Theoremresolution 27947 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.)
 |-  (
 ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 ch ) )  ->  ( ps  \/  ch )
 )
 
18.24  Mathbox for Alan Sare
 
18.24.1  Supplementary "adant" deductions
 
Theoremad4ant13 27948 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant14 27949 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant123 27950 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ta )  ->  th )
 
Theoremad4ant124 27951 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ta )  /\  ch )  ->  th )
 
Theoremad4ant134 27952 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ta )  /\  ps )  /\  ch )  ->  th )
 
Theoremad4ant23 27953 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant24 27954 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant234 27955 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ta 
 /\  ph )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant12 27956 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  ps )  /\  th )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant13 27957 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant14 27958 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant15 27959 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant23 27960 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant24 27961 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant25 27962 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant245 27963 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant234 27964 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant235 27965 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant123 27966 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ch )  /\  ta )  /\  et )  ->  th )
 
Theoremad5ant124 27967 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ta )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant125 27968 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ta )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant134 27969 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  ps )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant135 27970 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  ps )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant145 27971 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant1345 27972 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( ph  /\  et )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremad5ant2345 27973 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( et  /\  ph )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
18.24.2  Supplementary unification deductions
 
Theorembiimp 27974 Importation inference similar to imp 418, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ch )
 )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorembi2imp 27975 Importation inference similar to imp 418, except the both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theorembi3impb 27976 Similar to 3impb 1147 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi3impa 27977 Similar to 3impa 1146 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23impib 27978 3impib 1149 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ( ps 
 /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impib 27979 3impib 1149 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impib 27980 3impib 1149 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impia 27981 3impia 1148 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impia 27982 3impia 1148 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi33imp12 27983 3imp 1145 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23imp13 27984 3imp 1145 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13imp23 27985 3imp 1145 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi13imp2 27986 Similar to 3imp 1145 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch 
 <-> 
 th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi12imp3 27987 Similar to 3imp 1145 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi23imp1 27988 Similar to 3imp 1145 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123imp0 27989 Similar to 3imp 1145 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
18.24.3  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 27990 idn3 28121 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.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ch )
 ) )
 
Theoremee222 27991 e222 28142 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.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ( ps  ->  ta ) )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoremee3bir 27992 Right-biconditional form of e3 28255 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.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ta  <->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremee13 27993 e13 28266 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.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ps  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ch  ->  ( th  ->  et ) ) )
 
Theoremee121 27994 e121 28162 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee122 27995 e122 28159 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( ch  ->  ta ) )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee333 27996 e333 28251 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ze )
 ) )
 
Theoremee323 27997 e323 28284 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ze ) ) )
 
Theorem3ornot23 27998 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 28368. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
 
Theoremorbi1r 27999 orbi1 686 with order of disjuncts reversed. Derived from orbi1rVD 28369. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  \/  ph ) 
 <->  ( ch  \/  ps ) ) )
 
Theorembitr3 28000 Closed nested implication form of bitr3i 242. Derived automatically from bitr3VD 28370. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  ->  ( ps 
 <->  ch ) ) )
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