P. 448 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44701-44800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	in3 44701	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 44702	in3 44701 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 44703	The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 431 is the non-virtual deduction form of in3an 44703. (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 44704	The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 44704 is 3expia 1121. (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 44705	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 44706	Virtual deduction identity rule. simpr 484 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 44707	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 44708*	Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1928 is gen11 44708 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   ∀𝑥𝜓   )
Theorem	gen11nv 44709	Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1825 is gen11nv 44709 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 44710*	Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 44710 is alrimivv 1929 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 44711*	Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 44711 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   )
Theorem	gen21nv 44712	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   )
Theorem	gen31 44713*	Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 44713 is ggen31 44637 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 44714*	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 44715*	gen22 44714 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 44716	Existential Instantiation. Virtual deduction form of exlimexi 44616. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (   ∃𝑥𝜑   ,   𝜑   ▶   𝜓   )    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exinst01 44717	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 44718	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 44719	A Virtual deduction elimination rule. syl 17 is e1a 44719 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 44720	A Virtual deduction elimination rule. syl 17 is el1 44720 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 44721	Biconditional form of e1a 44719. sylib 218 is e1bi 44721 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 44722	Right biconditional form of e1a 44719. sylibr 234 is e1bir 44722 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 44723	A virtual deduction elimination rule. syl6 35 is e2 44723 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 44724	Biconditional form of e2 44723. imbitrdi 251 is e2bi 44724 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 44725	Right biconditional form of e2 44723. imbitrrdi 252 is e2bir 44725 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 44726	e223 44727 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 44727	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 44728	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 44729	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 44730	e220 44729 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 44731	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 44732	e202 44731 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 44733	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 44734	e022 44733 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 44735	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 44736	e002 44735 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 44737	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 44738	e020 44737 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 44739	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 44740	e200 44739 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 44741	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 44742	e221 44741 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 44743	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 44744	e212 44743 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 44745	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 44746	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 44747	e112 44746 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 44748	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 44749	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 44750	e211 44749 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 44751	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 44752	e210 44751 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 44753	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 44754	e201 44753 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 44755	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 44756	Virtual deduction rule e120 44755 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 44757	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 44758	e021 44757 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 44759	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 44760	e012 44759 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 44761	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 44762	e102 44761 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 44763	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 44764	Conjunction form of e22 44763. (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 44765	e22an 44764 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))
Theorem	e111 44766	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜑   ▶   𝜏   )
Theorem	e1111 44767	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 → (𝜏 → 𝜂))))    ⇒   ⊢ (   𝜑   ▶   𝜂   )
Theorem	e110 44768	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜑   ▶   𝜏   )
Theorem	ee110 44769	e110 44768 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ 𝜃    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	e101 44770	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜑   ▶   𝜏   )
Theorem	ee101 44771	e101 44770 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	e011 44772	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜓   ▶   𝜏   )
Theorem	ee011 44773	e011 44772 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	e100 44774	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜑   ▶   𝜏   )
Theorem	ee100 44775	e100 44774 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	e010 44776	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜓   ▶   𝜏   )
Theorem	ee010 44777	e010 44776 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ 𝜃    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	e001 44778	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜒   ▶   𝜏   )
Theorem	ee001 44779	e001 44778 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (𝜒 → 𝜃)    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	e11 44780	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 → (𝜒 → 𝜃))    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	e11an 44781	Conjunction form of e11 44780. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ▶   𝜒   )    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	ee11an 44782	e11an 44781 without virtual deductions. syl22anc 838 is also e11an 44781 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.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	e01 44783	A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (   𝜓   ▶   𝜃   )
Theorem	e01an 44784	Conjunction form of e01 44783. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   𝜓   ▶   𝜃   )
Theorem	ee01an 44785	e01an 44784 without virtual deductions. sylancr 587 is also a form of e01an 44784 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    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	e10 44786	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 → (𝜒 → 𝜃))    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	e10an 44787	Conjunction form of e10 44786. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	ee10an 44788	e10an 44787 without virtual deductions. sylancl 586 is also e10an 44787 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.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	e02 44789	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 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	e02an 44790	Conjunction form of e02 44789. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	ee02an 44791	e02an 44790 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜓 → (𝜒 → 𝜏))
Theorem	eel021old 44792	el021old 44793 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
Theorem	el021old 44793	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜃   )    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜏   )
Theorem	eel000cT 44794	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (⊤ → 𝜃)
Theorem	eel0TT 44795	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (⊤ → 𝜓)    &   ⊢ (⊤ → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	eelT00 44796	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	eelTTT 44797	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (⊤ → 𝜓)    &   ⊢ (⊤ → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	eelT11 44798	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	eelT1 44799	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Alan Sare, 23-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	eelT12 44800	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜓 ∧ 𝜃) → 𝜂)