P. 447 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44601-44700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfvd2impr 44601	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 44602	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 44603	The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 44603 is ex 412. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )    ⇒   ⊢ (   𝜑   ▶   (𝜓 → 𝜒)   )
Theorem	iin2 44604	in2 44602 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 44605	The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 415 is the non-virtual deduction form of in2an 44605. (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 44606	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 44607	in3 44606 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 44608	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 44608. (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 44609	The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 44609 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 44610	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 44611	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 44612	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 44613*	Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1924 is gen11 44613 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   ∀𝑥𝜓   )
Theorem	gen11nv 44614	Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1820 is gen11nv 44614 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 44615*	Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 44615 is alrimivv 1925 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 44616*	Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 44616 is alrimdv 1926 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 44617	Virtual deduction form of alrimdh 1860. (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 44618*	Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 44618 is ggen31 44542 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 44619*	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 44620*	gen22 44619 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 44621	Existential Instantiation. Virtual deduction form of exlimexi 44521. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (   ∃𝑥𝜑   ,   𝜑   ▶   𝜓   )    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exinst01 44622	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 44623	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 44624	A Virtual deduction elimination rule. syl 17 is e1a 44624 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 44625	A Virtual deduction elimination rule. syl 17 is el1 44625 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 44626	Biconditional form of e1a 44624. sylib 218 is e1bi 44626 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 44627	Right biconditional form of e1a 44624. sylibr 234 is e1bir 44627 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 44628	A virtual deduction elimination rule. syl6 35 is e2 44628 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 44629	Biconditional form of e2 44628. imbitrdi 251 is e2bi 44629 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 44630	Right biconditional form of e2 44628. imbitrrdi 252 is e2bir 44630 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 44631	e223 44632 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 44632	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 44633	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 44634	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 44635	e220 44634 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 44636	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 44637	e202 44636 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 44638	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 44639	e022 44638 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 44640	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 44641	e002 44640 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 44642	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 44643	e020 44642 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 44644	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 44645	e200 44644 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 44646	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 44647	e221 44646 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 44648	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 44649	e212 44648 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 44650	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 44651	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 44652	e112 44651 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 44653	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 44654	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 44655	e211 44654 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 44656	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 44657	e210 44656 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 44658	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 44659	e201 44658 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 44660	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 44661	Virtual deduction rule e120 44660 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 44662	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 44663	e021 44662 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 44664	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 44665	e012 44664 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 44666	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 44667	e102 44666 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 44668	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 44669	Conjunction form of e22 44668. (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 44670	e22an 44669 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 44671	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 44672	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 44673	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 44674	e110 44673 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 44675	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 44676	e101 44675 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 44677	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 44678	e011 44677 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 44679	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 44680	e100 44679 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 44681	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 44682	e010 44681 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 44683	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 44684	e001 44683 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 44685	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 44686	Conjunction form of e11 44685. (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 44687	e11an 44686 without virtual deductions. syl22anc 839 is also e11an 44686 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 44688	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 44689	Conjunction form of e01 44688. (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 44690	e01an 44689 without virtual deductions. sylancr 587 is also a form of e01an 44689 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 44691	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 44692	Conjunction form of e10 44691. (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 44693	e10an 44692 without virtual deductions. sylancl 586 is also e10an 44692 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 44694	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 44695	Conjunction form of e02 44694. (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 44696	e02an 44695 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 44697	el021old 44698 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 44698	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	eel132 44699	syl2an 596 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    &   ⊢ ((𝜓 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)
Theorem	eel000cT 44700	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)