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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-csc 27001* Define the cosecant function. We define it this way for cmpt 3974, which requires the form  ( x  e.  A  |->  B ). 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.)
 |-  csc  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 27002* Define the cotangent function. We define it this way for cmpt 3974, which requires the form  ( x  e.  A  |->  B ). 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.)
 |-  cot  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 27003 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 27004 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 27005 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 27006 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 27007 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 27008 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 27009 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 27010 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 27011 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 27012 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 27013 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 27014 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 27015 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 27016 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 27017 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 27018 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
 ) )
 
16.20.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 27019 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 3477 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
16.20.6  Not-member-of
 
TheoremAnelBC 27020 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 }
 
16.20.7  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 27024 and df-dp2 27023 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10004.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 27021 Constant used for decimal fraction constructor. See df-dp2 27023.
 class _ A B
 
Syntaxcdp 27022 Decimal point operator. See df-dp 27024.
 class  period
 
Definitiondf-dp2 27023 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10004. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 27024* 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 27025 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 27026 Define the value of the decimal point operator. See df-dp 27024. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 27027 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 27024. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 27028 Prove a simple equivalence involving the decimal point. See df-dp 27024 and dpcl 27027. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
16.20.8  Signum (sgn or sign) function
 
Syntaxcsgn 27029 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 27030 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 8751) instead of  RR so that it can accept  +oo and  -oo. Note that df-psgn 26581 defines the sign of a permutation, which is different. Value shown in sgnval 27031. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 , 
 0 ,  if ( x  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 27031 Value of Signum function. Pronounced "signum" . See df-sgn 27030. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 27032 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
 
Theoremsgnp 27033 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 27034 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
 |-  ( A  e.  RR+  ->  (sgn `  A )  =  1 )
 
Theoremsgn1 27035 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn `  1 )  =  1
 
Theoremsgnpnf 27036 Proof that the signum of  +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 +oo )  =  1
 
Theoremsgnn 27037 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 27038 Proof that the signum of  -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 -oo )  =  -u 1
 
16.20.9  Ceiling function
 
Syntaxccei 27039 Extend class notation to include the ceiling function.
 class
 
Definitiondf-ceiling 27040 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 10833 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 27041 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceilingcl 27042 Closure of the ceiling function; the real work is in ceicl 10833. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  e.  ZZ )
 
16.20.10  Logarithm laws generalized to an arbitrary base

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions. The way defined here supports two notations,  (logb `  <. B ,  X >. ) and  ( Blogb X ) where  B is the base and  X is the other parameter. An alternative would be to support the notational form  ( (logb `  B ) `  X
); that looks a little more like traditional notation, but is different than other 2-parameter functions. It's not obvious to me which is better, so here we try out one way as an experiment. Feedback and help welcome.

 
Syntaxclogb 27043 Extend class notation to include the logarithm generalized to an arbitrary base.
 class logb
 
Definitiondf-logb 27044* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as  (logb `  <. B ,  X >. ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". You could also use  ( Blogb X ), which looks like a less-common notation that some use where the base is a preceding superscript. Note: This definition doesn't prevent bases of 1 or 0; proofs may need to forbid them. (Contributed by David A. Wheeler, 21-Jan-2017.)
 |- logb  =  ( x  e.  CC ,  y  e.  CC  |->  ( ( log `  y )  /  ( log `  x ) ) )
 
Theoremlogbnfxval 27045 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the other operand here. Proof is similar to modval 10853. (Contributed by David A. Wheeler, 21-Jan-2017.)
 |-  (
 ( B  e.  CC  /\  X  e.  CC )  ->  ( Blogb X )  =  ( ( log `  X )  /  ( log `  B ) ) )
 
Theoremlogbval 27046 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the form logb <. B ,  X >. (Contributed by David A. Wheeler, 21-Jan-2017.)
 |-  (
 ( B  e.  CC  /\  X  e.  CC )  ->  (logb `  <. B ,  X >. )  =  ( ( log `  X )  /  ( log `  B ) ) )
 
Theoremreglogbcl 27047 General logarithm is a real number, given extended real numbers. Based on reglogcl 26141. (Contributed by David A. Wheeler, 21-Jan-2017.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+  /\  B  =/=  1 )  ->  (logb ` 
 <. B ,  A >. )  e.  RR )
 
16.20.11  Miscellaneous

Miscellaneous proofs.

 
Theorem2m1e1 27048 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9738. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  (
 2  -  1 )  =  1
 
Theorem5m4e1 27049 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 27050 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 27051 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 )
 )
 
16.21  Mathbox for Alan Sare
 
16.21.1  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 27052 idn3 27174 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 27053 e222 27195 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 27054 Right-biconditional form of e3 27299 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 27055 e13 27310 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 27056 e121 27215 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 27057 e122 27212 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 27058 e333 27295 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 27059 e323 27328 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 27060 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 27410. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
 
Theoremorbi1r 27061 orbi1 689 with order of disjuncts reversed. Derived from orbi1rVD 27411. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  \/  ph ) 
 <->  ( ch  \/  ps ) ) )
 
Theorembitr3 27062 Closed nested implication form of bitr3i 244. Derived automatically from bitr3VD 27412. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  ->  ( ps 
 <->  ch ) ) )
 
Theorem3orbi123 27063 pm4.39 846 with a 3-conjunct antecedent. This proof is 3orbi123VD 27413 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) ) )
 
Theoremsyl5imp 27064 Closed form of syl5 30. Derived automatically from syl5impVD 27426. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch )
 ) ) )
 
Theoremimpexp3a 27065 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.)
1::  |-  ( ( ( ps  /\  ch )  ->  th )  <->  ( ps  ->  ( ch  ->  th ) ) )
qed:1:  |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theoremcom3rgbi 27066 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.)
1::  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
2::  |-  ( ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
4::  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
5::  |-  ( ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
6:4,5:  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
qed:3,6:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  <->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  th )
 ) )  <->  ( ch  ->  (
 ph  ->  ( ps  ->  th ) ) ) )
 
Theoremimpexp3acom3r 27067 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.)
1::  |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
2::  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
qed:1,2:  |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 
Theoremee1111 27068 Non-virtual deduction form of e1111 27234. (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.
h1::  |-  ( ph  ->  ps )
h2::  |-  ( ph  ->  ch )
h3::  |-  ( ph  ->  th )
h4::  |-  ( ph  ->  ta )
h5::  |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) )
6:1,5:  |-  ( ph  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) )
7:6:  |-  ( ch  ->  ( ph  ->  ( th  ->  ( ta  ->  et ) ) ) )
8:2,7:  |-  ( ph  ->  ( ph  ->  ( th  ->  ( ta  ->  et ) ) ) )
9:8:  |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )
10:9:  |-  ( th  ->  ( ph  ->  ( ta  ->  et ) ) )
11:3,10:  |-  ( ph  ->  ( ph  ->  ( ta  ->  et ) ) )
12:11:  |-  ( ph  ->  ( ta  ->  et ) )
13:12:  |-  ( ta  ->  ( ph  ->  et ) )
14:4,13:  |-  ( ph  ->  ( ph  ->  et ) )
qed:14:  |-  ( ph  ->  et )
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) )   =>    |-  ( ph  ->  et )
 
Theorempm2.43bgbi 27069 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.
1::  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  ->  ( ph  ->  ( ph  ->  ( ps  ->  ch ) ) ) )
2::  |-  ( ( ph  ->  ( ph  ->  ( ps  ->  ch ) ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
4::  |-  ( ( ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
5:3,4:  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
6::  |-  ( ( ps  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
qed:5,6:  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  <->  ( ps  ->  ( ph  ->  ch ) ) )
 |-  (
 ( ph  ->  ( ps 
 ->  ( ph  ->  ch )
 ) )  <->  ( ps  ->  (
 ph  ->  ch ) ) )
 
Theorempm2.43cbi 27070 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.
1::  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ph  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) ) )
2::  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) )  )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) )
4::  |-  ( ( ps  ->  ( ph  ->  ( ch  ->  th ) ) )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
5:3,4:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
6::  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) ) )
qed:5,6:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  ( ph  ->  th ) ) ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 
Theoremee233 27071 Non-virtual deduction form of e233 27327. (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.
h1::  |-  ( ph  ->  ( ps  ->  ch ) )
h2::  |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )
h3::  |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
h4::  |-  ( ch  ->  ( ta  ->  ( et  ->  ze ) ) )
5:1,4:  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ze ) ) )  )
6:5:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) )  )
7:2,6:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
8:7:  |-  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) )
9:8:  |-  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) )  )
10:9:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( et  ->  ze ) ) )  )
11:10:  |-  ( et  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )  )
12:3,11:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
13:12:  |-  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) )
14:13:  |-  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )  )
qed:14:  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  et )
 ) )   &    |-  ( ch  ->  ( ta  ->  ( et  ->  ze ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ze )
 ) )
 
Theoremimbi12 27072 Implication form of imbi12i 318. imbi12 27072 is imbi12VD 27436 without virtual deductions and was automatically derived from imbi12VD 27436 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.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  th )
 ) ) )
 
Theoremimbi13 27073 Join three logical equivalences to form equivalence of implications. imbi13 27073 is imbi13VD 27437 without virtual deductions and was automatically derived from imbi13VD 27437 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.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ta  <->  et )  ->  (
 ( ph  ->  ( ch 
 ->  ta ) )  <->  ( ps  ->  ( th  ->  et )
 ) ) ) ) )
 
Theoremee33 27074 Non-virtual deduction form of e33 27296. (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.
h1::  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
h2::  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
h3::  |-  ( th  ->  ( ta  ->  et ) )
4:1,3:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
5:4:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
6:2,5:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7:6:  |-  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
8:7:  |-  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
qed:8:  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 
Theoremcon5 27075 Bi-conditional contraposition variation. This proof is con5VD 27463 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  -.  ps )  ->  ( -.  ph  ->  ps )
 )
 
Theoremcon5i 27076 Inference form of con5 27075. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  -.  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremexlimexi 27077 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.)
 |-  ( ps  ->  A. x ps )   &    |-  ( E. x ph  ->  ( ph  ->  ps ) )   =>    |-  ( E. x ph 
 ->  ps )
 
Theoremsb5ALT 27078* Equivalence for substitution. Alternate proof of sb5 1993. This proof is sb5ALTVD 27476 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremeexinst01 27079 exinst01 27184 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  E. x ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  ( ph  ->  ch )
 
Theoremeexinst11 27080 exinst11 27185 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  ( ph  ->  ch )
 
Theoremvk15.4j 27081 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 27081 is vk15.4jVD 27477 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( E. x  -.  ph  /\ 
 E. x ( ps 
 /\  -.  ch )
 )   &    |-  ( A. x ch  ->  -.  E. x ( th  /\  ta )
 )   &    |- 
 -.  A. x ( ta 
 ->  ph )   =>    |-  ( -.  E. x  -.  th  ->  -.  A. x ps )
 
Theoremnotnot2ALT 27082 Converse of double negation. Alternate proof of notnot2 106. This proof is notnot2ALTVD 27478 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  -.  ph  ->  ph )
 
Theoremcon3ALT 27083 Contraposition. Alternate proof of con3 128. This proof is con3ALTVD 27479 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
 
Theoremssralv2 27084* Quantification restricted to a subclass for two quantifiers. ssralv 3158 for two quantifiers. The proof of ssralv2 27084 was automatically generated by minimizing the automatically translated proof of ssralv2VD 27429. The automatic translation is by the tools program translatewithout_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  C  C_  D )  ->  ( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  C  ph ) )
 
Theoremsbc3org 27085 sbcorg 2966 with a 3-disjuncts. This proof is sbc3orgVD 27414 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ].
 ch ) ) )
 
Theoremalrim3con13v 27086* Closed form of alrimi 1706 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 27415 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps 
 /\  ph  /\  ch )
 ) )
 
Theoremra4sbc2 27087* ra4sbc 2999 with two quantifying variables. This proof is ra4sbc2VD 27418 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  ( C  e.  D  ->  (
 A. x  e.  B  A. y  e.  D  ph  -> 
 [. C  /  y ]. [. A  /  x ].
 ph ) ) )
 
Theoremsbcoreleleq 27088* Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 27422. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  V  ->  (
 [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
 ) 
 <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A )
 ) )
 
Theoremtratrb 27089* 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 27424. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) 
 ->  Tr  B )
 
Theorem3ax5 27090 ax-5 1533 for a 3 element left-nested implication. Derived automatically from 3ax5VD 27425. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  ( ps  ->  ch )
 )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
 
TheoremordelordALT 27091 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4307 using the Axiom of Regularity indirectly through dford2 7205. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that  _E  Fr  A because this is inferred by the Axiom of Regularity. ordelordALT 27091 is ordelordALTVD 27430 without virtual deductions and was automatically derived from ordelordALTVD 27430 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Ord  A  /\  B  e.  A )  ->  Ord  B )
 
Theoremsbcim2g 27092 Distribution of class substitution over a left-nested implication. Similar to sbcimg 2962. sbcim2g 27092 is sbcim2gVD 27438 without virtual deductions and was automatically derived from sbcim2gVD 27438 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.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
 [. A  /  x ].
 ps  ->  [. A  /  x ].
 ch ) ) ) )
 
Theoremsbcbi 27093 Implication form of sbcbiiOLD 2977. sbcbi 27093 is sbcbiVD 27439 without virtual deductions and was automatically derived from sbcbiVD 27439 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.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) ) )
 
Theoremtrsbc 27094* Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. trsbc 27094 is trsbcVD 27440 without virtual deductions and was automatically derived from trsbcVD 27440 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.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ].
 Tr  x  <->  Tr  A ) )
 
TheoremtruniALT 27095* The union of a class of transitive sets is transitive. Alternate proof of truni 4024. truniALT 27095 is truniALTVD 27441 without virtual deductions and was automatically derived from truniALTVD 27441 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.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
 
Theoremsbcss 27096 Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssVD 27446. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )
 
TheoremonfrALTlem5 27097* Lemma for onfrALT 27104. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [. ( a  i^i  x )  /  b ]. (
 ( b  C_  (
 a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x ) 
 C_  ( a  i^i 
 x )  /\  (
 a  i^i  x )  =/= 
 (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
 
TheoremonfrALTlem4 27098* Lemma for onfrALT 27104. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
 
TheoremonfrALTlem3 27099* Lemma for onfrALT 27104. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( a  C_  On  /\  a  =/=  (/) )  ->  ( ( x  e.  a  /\  -.  (
 a  i^i  x )  =  (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
 
Theoremggen31 27100* gen31 27180 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  A. x th ) ) )
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