P. 423 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42201-42300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	idn1 42201	Virtual deduction identity rule which is id 22 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜑   )
Theorem	dfvd1imp 42202	Left-to-right part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((   𝜑   ▶   𝜓   ) → (𝜑 → 𝜓))
Theorem	dfvd1impr 42203	Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (   𝜑   ▶   𝜓   ))
Syntax	wvd2 42204	Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.)
wff (   𝜑   ,   𝜓   ▶   𝜒   )
Definition	df-vd2 42205	Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)
⊢ ((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
Theorem	dfvd2 42206	Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ (𝜑 → (𝜓 → 𝜒)))
Syntax	wvhc2 42207	Syntax for a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)
wff (   𝜑   ,   𝜓   )
Definition	df-vhc2 42208	Definition of a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)
⊢ ((   𝜑   ,   𝜓   ) ↔ (𝜑 ∧ 𝜓))
Theorem	dfvd2an 42209	Definition of a 2-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((   (   𝜑   ,   𝜓   )   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
Theorem	dfvd2ani 42210	Inference form of dfvd2an 42209. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	dfvd2anir 42211	Right-to-left inference form of dfvd2an 42209. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )
Theorem	dfvd2i 42212	Inference form of dfvd2 42206. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	dfvd2ir 42213	Right-to-left inference form of dfvd2 42206. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
Syntax	wvd3 42214	Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.)
wff (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
Syntax	wvhc3 42215	Syntax for a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)
wff (   𝜑   ,   𝜓   ,   𝜒   )
Definition	df-vhc3 42216	Definition of a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)
⊢ ((   𝜑   ,   𝜓   ,   𝜒   ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
Definition	df-vd3 42217	Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)
⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
Theorem	dfvd3 42218	Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
Theorem	dfvd3i 42219	Inference form of dfvd3 42218. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
Theorem	dfvd3ir 42220	Right-to-left inference form of dfvd3 42218. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
Theorem	dfvd3an 42221	Definition of a 3-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
Theorem	dfvd3ani 42222	Inference form of dfvd3an 42221. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   )    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	dfvd3anir 42223	Right-to-left inference form of dfvd3an 42221. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   )
Theorem	vd01 42224	A virtual hypothesis virtually infers a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ (   𝜓   ▶   𝜑   )
Theorem	vd02 42225	Two virtual hypotheses virtually infer a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜑   )
Theorem	vd03 42226	A theorem is virtually inferred by the 3 virtual hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜑   )
Theorem	vd12 42227	A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and an additional hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜓   )
Theorem	vd13 42228	A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and a two additional hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜓   )
Theorem	vd23 42229	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   )
Theorem	dfvd2imp 42230	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 → (𝜓 → 𝜒)))
Theorem	dfvd2impr 42231	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 → (𝜓 → 𝜒)) → (   𝜑   ,   𝜓   ▶   𝜒   ))
Theorem	in2 42232	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   ▶   (𝜓 → 𝜒)   )
Theorem	int2 42233	The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 42233 is ex 413. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )    ⇒   ⊢ (   𝜑   ▶   (𝜓 → 𝜒)   )
Theorem	iin2 42234	in2 42232 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	in2an 42235	The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 416 is the non-virtual deduction form of in2an 42235. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   (𝜓 ∧ 𝜒)   ▶   𝜃   )    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   (𝜒 → 𝜃)   )
Theorem	in3 42236	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   ▶   (𝜒 → 𝜃)   )
Theorem	iin3 42237	in3 42236 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 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
Theorem	in3an 42238	The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 432 is the non-virtual deduction form of in3an 42238. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   (𝜒 ∧ 𝜃)   ▶   𝜏   )    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   (𝜃 → 𝜏)   )
Theorem	int3 42239	The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 42239 is 3expia 1120. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   )    ⇒   ⊢ (   (   𝜑   ,   𝜓   )   ▶   (𝜒 → 𝜃)   )
Theorem	idn2 42240	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   )
Theorem	iden2 42241	Virtual deduction identity rule. simpr 485 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜓   )
Theorem	idn3 42242	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   )
Theorem	gen11 42243*	Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1931 is gen11 42243 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   ∀𝑥𝜓   )
Theorem	gen11nv 42244	Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1827 is gen11nv 42244 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   ∀𝑥𝜓   )
Theorem	gen12 42245*	Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 42245 is alrimivv 1932 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   ∀𝑥∀𝑦𝜓   )
Theorem	gen21 42246*	Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 42246 is alrimdv 1933 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )
Theorem	gen21nv 42247	Virtual deduction form of alrimdh 1867. (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   )
Theorem	gen31 42248*	Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 42248 is ggen31 42172 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   )
Theorem	gen22 42249*	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   ▶   ∀𝑥∀𝑦𝜒   )
Theorem	ggen22 42250*	gen22 42249 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	exinst 42251	Existential Instantiation. Virtual deduction form of exlimexi 42151. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (   ∃𝑥𝜑   ,   𝜑   ▶   𝜓   )    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exinst01 42252	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   )
Theorem	exinst11 42253	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   )
Theorem	e1a 42254	A Virtual deduction elimination rule. syl 17 is e1a 42254 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (   𝜑   ▶   𝜒   )
Theorem	el1 42255	A Virtual deduction elimination rule. syl 17 is el1 42255 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (   𝜑   ▶   𝜒   )
Theorem	e1bi 42256	Biconditional form of e1a 42254. sylib 217 is e1bi 42256 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (   𝜑   ▶   𝜒   )
Theorem	e1bir 42257	Right biconditional form of e1a 42254. sylibr 233 is e1bir 42257 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (   𝜑   ▶   𝜒   )
Theorem	e2 42258	A virtual deduction elimination rule. syl6 35 is e2 42258 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
Theorem	e2bi 42259	Biconditional form of e2 42258. syl6ib 250 is e2bi 42259 without virtual deductions. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
Theorem	e2bir 42260	Right biconditional form of e2 42258. syl6ibr 251 is e2bir 42260 without virtual deductions. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
Theorem	ee223 42261	e223 42262 without virtual deductions. (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 → (𝜂 → 𝜁)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜁)))
Theorem	e223 42262	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 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )
Theorem	e222 42263	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	e220 42264	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee220 42265	e220 42264 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ 𝜏    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e202 42266	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee202 42267	e202 42266 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ 𝜃    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e022 42268	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee022 42269	e022 42268 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ (𝜓 → (𝜒 → 𝜏))    &   ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜂))
Theorem	e002 42270	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜒   ,   𝜃   ▶   𝜂   )
Theorem	ee002 42271	e002 42270 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (𝜒 → (𝜃 → 𝜏))    &   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜒 → (𝜃 → 𝜂))
Theorem	e020 42272	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee020 42273	e020 42272 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ 𝜏    &   ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜂))
Theorem	e200 42274	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee200 42275	e200 42274 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ 𝜃    &   ⊢ 𝜏    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e221 42276	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee221 42277	e221 42276 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e212 42278	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee212 42279	e212 42278 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e122 42280	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜂   )
Theorem	e112 42281	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜃   ▶   𝜂   )
Theorem	ee112 42282	e112 42281 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜃 → 𝜏))    &   ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜃 → 𝜂))
Theorem	e121 42283	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜂   )
Theorem	e211 42284	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee211 42285	e211 42284 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e210 42286	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee210 42287	e210 42286 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜃)    &   ⊢ 𝜏    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e201 42288	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )
Theorem	ee201 42289	e201 42288 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ 𝜃    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))
Theorem	e120 42290	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜂   )
Theorem	ee120 42291	Virtual deduction rule e120 42290 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ 𝜏    &   ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜂))
Theorem	e021 42292	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee021 42293	e021 42292 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ (𝜓 → 𝜏)    &   ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜂))
Theorem	e012 42294	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜓   ,   𝜃   ▶   𝜂   )
Theorem	ee012 42295	e012 42294 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → (𝜃 → 𝜏))    &   ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜓 → (𝜃 → 𝜂))
Theorem	e102 42296	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 → (𝜏 → 𝜂)))    ⇒   ⊢ (   𝜑   ,   𝜃   ▶   𝜂   )
Theorem	ee102 42297	e102 42296 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ (𝜑 → (𝜃 → 𝜏))    &   ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜃 → 𝜂))
Theorem	e22 42298	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 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
Theorem	e22an 42299	Conjunction form of e22 42298. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
Theorem	ee22an 42300	e22an 42299 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜏))