P. 444 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44301-44400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	exinst01 44301	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 44302	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 44303	A Virtual deduction elimination rule. syl 17 is e1a 44303 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 44304	A Virtual deduction elimination rule. syl 17 is el1 44304 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 44305	Biconditional form of e1a 44303. sylib 217 is e1bi 44305 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 44306	Right biconditional form of e1a 44303. sylibr 233 is e1bir 44306 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 44307	A virtual deduction elimination rule. syl6 35 is e2 44307 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 44308	Biconditional form of e2 44307. imbitrdi 250 is e2bi 44308 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 44309	Right biconditional form of e2 44307. imbitrrdi 251 is e2bir 44309 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 44310	e223 44311 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 44311	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 44312	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 44313	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 44314	e220 44313 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 44315	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 44316	e202 44315 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 44317	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 44318	e022 44317 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 44319	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 44320	e002 44319 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 44321	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 44322	e020 44321 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 44323	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 44324	e200 44323 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 44325	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 44326	e221 44325 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 44327	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 44328	e212 44327 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 44329	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 44330	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 44331	e112 44330 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 44332	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 44333	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 44334	e211 44333 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 44335	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 44336	e210 44335 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 44337	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 44338	e201 44337 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 44339	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 44340	Virtual deduction rule e120 44339 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 44341	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 44342	e021 44341 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 44343	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 44344	e012 44343 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 44345	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 44346	e102 44345 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 44347	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 44348	Conjunction form of e22 44347. (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 44349	e22an 44348 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 44350	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 44351	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 44352	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 44353	e110 44352 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 44354	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 44355	e101 44354 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 44356	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 44357	e011 44356 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 44358	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 44359	e100 44358 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 44360	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 44361	e010 44360 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 44362	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 44363	e001 44362 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 44364	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 44365	Conjunction form of e11 44364. (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 44366	e11an 44365 without virtual deductions. syl22anc 837 is also e11an 44365 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 44367	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 44368	Conjunction form of e01 44367. (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 44369	e01an 44368 without virtual deductions. sylancr 585 is also a form of e01an 44368 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 44370	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 44371	Conjunction form of e10 44370. (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 44372	e10an 44371 without virtual deductions. sylancl 584 is also e10an 44371 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 44373	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 44374	Conjunction form of e02 44373. (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 44375	e02an 44374 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 44376	el021old 44377 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 44377	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 44378	syl2an 594 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 44379	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 44380	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 44381	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 44382	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 44383	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 44384	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 44385	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)
Theorem	eelTT1 44386	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (⊤ → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	eelT01 44387	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ 𝜓    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	eel0T1 44388	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (⊤ → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	eel12131 44389	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)
Theorem	eel2131 44390	syl2an 594 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜂)
Theorem	eel3132 44391	syl2an 594 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜃 ∧ 𝜓) → 𝜏)    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜂)
Theorem	eel0321old 44392	el0321old 44393 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜂)
Theorem	el0321old 44393	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜏   )    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜂   )
Theorem	eel2122old 44394	el2122old 44395 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ (𝜓 → 𝜏)    &   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
Theorem	el2122old 44395	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )    &   ⊢ (   𝜓   ▶   𝜃   )    &   ⊢ (   𝜓   ▶   𝜏   )    &   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜂   )
Theorem	eel0000 44396	Elimination rule similar to mp4an 691, except with a left-nested conjunction unification theorem. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ 𝜏
Theorem	eel00001 44397	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ (𝜏 → 𝜂)    &   ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜁)    ⇒   ⊢ (𝜏 → 𝜁)
Theorem	eel00000 44398	Elimination rule similar eel0000 44396, except with five hpothesis steps. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ 𝜏    &   ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ 𝜂
Theorem	eel11111 44399	Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1379 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)    ⇒   ⊢ (𝜑 → 𝜁)
Theorem	e12 44400	A virtual deduction elimination rule (see sylsyld 61). (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜏   )