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Theorem List for Metamath Proof Explorer - 41301-41400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvd23 41301 A virtual deduction with 2 virtual hypotheses virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same 2 virtual hypotheses and a third hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )

Theoremdfvd2imp 41302 The virtual deduction form of a 2-antecedent nested implication implies the 2-antecedent nested implication. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((   𝜑   ,   𝜓   ▶   𝜒   ) → (𝜑 → (𝜓𝜒)))

Theoremdfvd2impr 41303 A 2-antecedent nested implication implies its virtual deduction form. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (   𝜑   ,   𝜓   ▶   𝜒   ))

Theoremin2 41304 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ▶   (𝜓𝜒)   )

Theoremint2 41305 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 41305 is ex 416. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,   𝜓   )   ▶   𝜒   )       (   𝜑   ▶   (𝜓𝜒)   )

Theoremiin2 41306 in2 41304 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theoremin2an 41307 The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 419 is the non-virtual deduction form of in2an 41307. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   (𝜓𝜒)   ▶   𝜃   )       (   𝜑   ,   𝜓   ▶   (𝜒𝜃)   )

Theoremin3 41308 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )       (   𝜑   ,   𝜓   ▶   (𝜒𝜃)   )

Theoremiin3 41309 in3 41308 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremin3an 41310 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 435 is the non-virtual deduction form of in3an 41310. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   (𝜒𝜃)   ▶   𝜏   )       (   𝜑   ,   𝜓   ,   𝜒   ▶   (𝜃𝜏)   )

Theoremint3 41311 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 41311 is 3expia 1118. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   )       (   (   𝜑   ,   𝜓   )   ▶   (𝜒𝜃)   )

Theoremidn2 41312 Virtual deduction identity rule which is idd 24 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜓   )

Theoremiden2 41313 Virtual deduction identity rule. simpr 488 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,   𝜓   )   ▶   𝜓   )

Theoremidn3 41314 Virtual deduction identity rule for three virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜒   )

Theoremgen11 41315* Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1928 is gen11 41315 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )       (   𝜑   ▶   𝑥𝜓   )

Theoremgen11nv 41316 Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1825 is gen11nv 41316 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (   𝜑   ▶   𝜓   )       (   𝜑   ▶   𝑥𝜓   )

Theoremgen12 41317* Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 41317 is alrimivv 1929 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )       (   𝜑   ▶   𝑥𝑦𝜓   )

Theoremgen21 41318* Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 41318 is alrimdv 1930 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ▶   𝑥𝜒   )

Theoremgen21nv 41319 Virtual deduction form of alrimdh 1864. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ▶   𝑥𝜒   )

Theoremgen31 41320* Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 41320 is ggen31 41244 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝑥𝜃   )

Theoremgen22 41321* Virtual deduction generalizing rule for two quantifying variables and two virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ▶   𝑥𝑦𝜒   )

Theoremggen22 41322* gen22 41321 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝑦𝜒))

Theoremexinst 41323 Existential Instantiation. Virtual deduction form of exlimexi 41223. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (   𝑥𝜑   ,   𝜑   ▶   𝜓   )       (∃𝑥𝜑𝜓)

Theoremexinst01 41324 Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (   𝜑   ▶   𝜒   )

Theoremexinst11 41325 Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝑥𝜓   )    &   (   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (   𝜑   ▶   𝜒   )

Theoreme1a 41326 A Virtual deduction elimination rule. syl 17 is e1a 41326 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜓𝜒)       (   𝜑   ▶   𝜒   )

Theoremel1 41327 A Virtual deduction elimination rule. syl 17 is el1 41327 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜓𝜒)       (   𝜑   ▶   𝜒   )

Theoreme1bi 41328 Biconditional form of e1a 41326. sylib 221 is e1bi 41328 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜓𝜒)       (   𝜑   ▶   𝜒   )

Theoreme1bir 41329 Right biconditional form of e1a 41326. sylibr 237 is e1bir 41329 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜒𝜓)       (   𝜑   ▶   𝜒   )

Theoreme2 41330 A virtual deduction elimination rule. syl6 35 is e2 41330 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜒𝜃)       (   𝜑   ,   𝜓   ▶   𝜃   )

Theoreme2bi 41331 Biconditional form of e2 41330. syl6ib 254 is e2bi 41331 without virtual deductions. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜒𝜃)       (   𝜑   ,   𝜓   ▶   𝜃   )

Theoreme2bir 41332 Right biconditional form of e2 41330. syl6ibr 255 is e2bir 41332 without virtual deductions. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜃𝜒)       (   𝜑   ,   𝜓   ▶   𝜃   )

Theoremee223 41333 e223 41334 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓 → (𝜏𝜂)))    &   (𝜒 → (𝜃 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜏𝜁)))

Theoreme223 41334 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )    &   (𝜒 → (𝜃 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )

Theoreme222 41335 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoreme220 41336 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee220 41337 e220 41336 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme202 41338 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   𝜃    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee202 41339 e202 41338 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme022 41340 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜓   ,   𝜒   ▶   𝜏   )    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee022 41341 e022 41340 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   (𝜓 → (𝜒𝜏))    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (𝜓 → (𝜒𝜂))

Theoreme002 41342 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (   𝜒   ,   𝜃   ▶   𝜏   )    &   (𝜑 → (𝜓 → (𝜏𝜂)))       (   𝜒   ,   𝜃   ▶   𝜂   )

Theoremee002 41343 e002 41342 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (𝜒 → (𝜃𝜏))    &   (𝜑 → (𝜓 → (𝜏𝜂)))       (𝜒 → (𝜃𝜂))

Theoreme020 41344 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ▶   𝜃   )    &   𝜏    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee020 41345 e020 41344 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   𝜏    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (𝜓 → (𝜒𝜂))

Theoreme200 41346 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   𝜃    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee200 41347 e200 41346 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme221 41348 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (   𝜑   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee221 41349 e221 41348 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑𝜏)    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme212 41350 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee212 41351 e212 41350 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme122 41352 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜒   ▶   𝜏   )    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜒   ▶   𝜂   )

Theoreme112 41353 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (   𝜑   ,   𝜃   ▶   𝜏   )    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (   𝜑   ,   𝜃   ▶   𝜂   )

Theoremee112 41354 e112 41353 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → (𝜃𝜏))    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (𝜑 → (𝜃𝜂))

Theoreme121 41355 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ▶   𝜏   )    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜒   ▶   𝜂   )

Theoreme211 41356 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   (   𝜑   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee211 41357 e211 41356 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme210 41358 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee210 41359 e210 41358 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme201 41360 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   𝜃    &   (   𝜑   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee201 41361 e201 41360 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   (𝜑𝜏)    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme120 41362 A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   𝜏    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜒   ▶   𝜂   )

Theoremee120 41363 Virtual deduction rule e120 41362 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   𝜏    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))

Theoreme021 41364 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜓   ▶   𝜏   )    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee021 41365 e021 41364 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   (𝜓𝜏)    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (𝜓 → (𝜒𝜂))

Theoreme012 41366 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   (   𝜓   ,   𝜃   ▶   𝜏   )    &   (𝜑 → (𝜒 → (𝜏𝜂)))       (   𝜓   ,   𝜃   ▶   𝜂   )

Theoremee012 41367 e012 41366 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   (𝜓 → (𝜃𝜏))    &   (𝜑 → (𝜒 → (𝜏𝜂)))       (𝜓 → (𝜃𝜂))

Theoreme102 41368 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   (   𝜑   ,   𝜃   ▶   𝜏   )    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (   𝜑   ,   𝜃   ▶   𝜂   )

Theoremee102 41369 e102 41368 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   (𝜑 → (𝜃𝜏))    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (𝜑 → (𝜃𝜂))

Theoreme22 41370 A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (𝜒 → (𝜃𝜏))       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoreme22an 41371 Conjunction form of e22 41370. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   ((𝜒𝜃) → 𝜏)       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoremee22an 41372 e22an 41371 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   ((𝜒𝜃) → 𝜏)       (𝜑 → (𝜓𝜏))

Theoreme111 41373 A virtual deduction elimination rule (see syl3c 66). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoreme1111 41374 A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   (   𝜑   ▶   𝜏   )    &   (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))       (   𝜑   ▶   𝜂   )

Theoreme110 41375 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   𝜃    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoremee110 41376 e110 41375 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑𝜒)    &   𝜃    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoreme101 41377 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   (   𝜑   ▶   𝜃   )    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoremee101 41378 e101 41377 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   (𝜑𝜃)    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoreme011 41379 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   (   𝜓   ▶   𝜃   )    &   (𝜑 → (𝜒 → (𝜃𝜏)))       (   𝜓   ▶   𝜏   )

Theoremee011 41380 e011 41379 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   (𝜓𝜃)    &   (𝜑 → (𝜒 → (𝜃𝜏)))       (𝜓𝜏)

Theoreme100 41381 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   𝜃    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoremee100 41382 e100 41381 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   𝜃    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoreme010 41383 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   𝜃    &   (𝜑 → (𝜒 → (𝜃𝜏)))       (   𝜓   ▶   𝜏   )

Theoremee010 41384 e010 41383 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   𝜃    &   (𝜑 → (𝜒 → (𝜃𝜏)))       (𝜓𝜏)

Theoreme001 41385 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (   𝜒   ▶   𝜃   )    &   (𝜑 → (𝜓 → (𝜃𝜏)))       (   𝜒   ▶   𝜏   )

Theoremee001 41386 e001 41385 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (𝜒𝜃)    &   (𝜑 → (𝜓 → (𝜃𝜏)))       (𝜒𝜏)

Theoreme11 41387 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (𝜓 → (𝜒𝜃))       (   𝜑   ▶   𝜃   )

Theoreme11an 41388 Conjunction form of e11 41387. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )

Theoremee11an 41389 e11an 41388 without virtual deductions. syl22anc 837 is also e11an 41388 without virtual deductions, exept with a different order of hypotheses. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoreme01 41390 A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   (𝜑 → (𝜒𝜃))       (   𝜓   ▶   𝜃   )

Theoreme01an 41391 Conjunction form of e01 41390. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   ((𝜑𝜒) → 𝜃)       (   𝜓   ▶   𝜃   )

Theoremee01an 41392 e01an 41391 without virtual deductions. sylancr 590 is also a form of e01an 41391 without virtual deduction, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       (𝜓𝜃)

Theoreme10 41393 A virtual deduction elimination rule (see mpisyl 21). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   (𝜓 → (𝜒𝜃))       (   𝜑   ▶   𝜃   )

Theoreme10an 41394 Conjunction form of e10 41393. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )

Theoremee10an 41395 e10an 41394 without virtual deductions. sylancl 589 is also e10an 41394 without virtual deductions, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoreme02 41396 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ▶   𝜃   )    &   (𝜑 → (𝜃𝜏))       (   𝜓   ,   𝜒   ▶   𝜏   )

Theoreme02an 41397 Conjunction form of e02 41396. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ▶   𝜃   )    &   ((𝜑𝜃) → 𝜏)       (   𝜓   ,   𝜒   ▶   𝜏   )

Theoremee02an 41398 e02an 41397 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   ((𝜑𝜃) → 𝜏)       (𝜓 → (𝜒𝜏))

Theoremeel021old 41399 el021old 41400 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   ((𝜓𝜒) → 𝜃)    &   ((𝜑𝜃) → 𝜏)       ((𝜓𝜒) → 𝜏)

Theoremel021old 41400 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   (   𝜓   ,   𝜒   )   ▶   𝜃   )    &   ((𝜑𝜃) → 𝜏)       (   (   𝜓   ,   𝜒   )   ▶   𝜏   )

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45326
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