P. 400 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

ee13an 39901

e13an 39900 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    &   ⊢ ((𝜓 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜂)))

Theorem

e31 39902

A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ▶   𝜏   )    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theorem

ee31 39903

e31 39902 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Theorem

e31an 39904

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   )    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theorem

ee31an 39905

e31an 39904 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → 𝜏)    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Theorem

e23 39906

A virtual deduction elimination rule (see syl10 79). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   ⊢ (𝜒 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )

Theorem

e23an 39907

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 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )

Theorem

ee23an 39908

e23an 39907 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))

Theorem

e32 39909

A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theorem

ee32 39910

e32 39909 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Theorem

e32an 39911

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   )    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theorem

ee32an 39912

e33an 39886 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Theorem

e123 39913

A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜂   )    &   ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )

Theorem

ee123 39914

e123 39913 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜂)))    &   ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))    ⇒   ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜁)))

Theorem

el123 39915

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   ▶   𝜂   )    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )

Theorem

e233 39916

A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )    &   ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )

Theorem

e323 39917

A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )    &   ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Theorem

e000 39918

A virtual deduction elimination rule. The non-virtual deduction form of e000 39918 is the virtual deduction form. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ 𝜃

Theorem

e00 39919

Elimination rule identical to mp2 9. The non-virtual deduction form is the virtual deduction form, which is mp2 9. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ 𝜒

Theorem

e00an 39920

Elimination rule identical to mp2an 682. The non-virtual deduction form is the virtual deduction form, which is mp2an 682. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒

Theorem

eel00cT 39921

An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (⊤ → 𝜒)

Theorem

eelTT 39922

An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (⊤ → 𝜑)    &   ⊢ (⊤ → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒

Theorem

e0a 39923

Elimination rule identical to ax-mp 5. The non-virtual deduction form is the virtual deduction form, which is ax-mp 5. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓

Theorem

eelT 39924

An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (⊤ → 𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓

Theorem

eel0cT 39925

An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (⊤ → 𝜓)

Theorem

eelT0 39926

An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (⊤ → 𝜑)    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒

Theorem

e0bi 39927

Elimination rule identical to mpbi 222. The non-virtual deduction form is the virtual deduction form, which is mpbi 222. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓

Theorem

e0bir 39928

Elimination rule identical to mpbir 223. The non-virtual deduction form is the virtual deduction form, which is mpbir 223. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ 𝜓

Theorem

uun0.1 39929

Convention notation form of un0.1 39930. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((⊤ ∧ 𝜓) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)

Theorem

un0.1 39930

⊤ is the constant true, a tautology (see df-tru 1605). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   ⊤   ▶   𝜑   )    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )    ⇒   ⊢ (   𝜓   ▶   𝜃   )

Theorem

uunT1 39931

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accommodate a possible future version of df-tru 1605. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunT1p1 39932

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ ⊤) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunT21 39933

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uun121 39934

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uun121p1 39935

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uun132 39936

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun132p1 39937

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

anabss7p1 39938

A deduction unionizing a non-unionized collection of virtual hypotheses. This would have been named uun221 if the 0th permutation did not exist in set.mm as anabss7 663. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Theorem

un10 39939

A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   (   𝜑   ,   ⊤   )   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   𝜓   )

Theorem

un01 39940

A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   (   ⊤   ,   𝜑   )   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   𝜓   )

Theorem

un2122 39941

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uun2131 39942

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun2131p1 39943

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uunTT1 39944

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ ⊤ ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunTT1p1 39945

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunTT1p2 39946

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ ⊤ ∧ ⊤) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunT11 39947

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunT11p1 39948

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunT11p2 39949

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜑 ∧ ⊤) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

uunT12 39950

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ 𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uunT12p1 39951

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uunT12p2 39952

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uunT12p3 39953

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uunT12p4 39954

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜓 ∧ ⊤) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uunT12p5 39955

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uun111 39956

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

3anidm12p1 39957

A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1491 denotes the deduction which would have been named uun112 if it did not pre-exist in set.mm. This second permutation's name is based on this pre-existing name. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

3anidm12p2 39958

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜑 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Theorem

uun123 39959

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun123p1 39960

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun123p2 39961

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun123p3 39962

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun123p4 39963

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Theorem

uun2221 39964

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Theorem

uun2221p1 39965

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Theorem

uun2221p2 39966

A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Theorem

3impdirp1 39967

A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir 1413. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)

Theorem

3impcombi 39968

A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)

⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)

20.32.6  Theorems proved using Virtual Deduction

Theorem

trsspwALT 39969

Virtual deduction proof of the left-to-right implication of dftr4 4992. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4992 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Theorem

trsspwALT2 39970

Virtual deduction proof of trsspwALT 39969. This proof is the same as the proof of trsspwALT 39969 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Theorem

trsspwALT3 39971

Short predicate calculus proof of the left-to-right implication of dftr4 4992. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 39970, which is the virtual deduction proof trsspwALT 39969 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Theorem

sspwtr 39972

Virtual deduction proof of the right-to-left implication of dftr4 4992. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 39972 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Theorem

sspwtrALT 39973

Virtual deduction proof of sspwtr 39972. This proof is the same as the proof of sspwtr 39972 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Theorem

sspwtrALT2 39974

Short predicate calculus proof of the right-to-left implication of dftr4 4992. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 39973, which is the virtual deduction proof sspwtr 39972 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Theorem

pwtrVD 39975

Virtual deduction proof of pwtr 5153; see pwtrrVD 39976 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → Tr 𝒫 𝐴)

Theorem

pwtrrVD 39976

Virtual deduction proof of pwtr 5153; see pwtrVD 39975 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝒫 𝐴 → Tr 𝐴)

Theorem

suctrALT 39977

The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 39977 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 6059 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → Tr suc 𝐴)

Theorem

snssiALTVD 39978

Virtual deduction proof of snssiALT 39979. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Theorem

snssiALT 39979

If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4570. This theorem was automatically generated from snssiALTVD 39978 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Theorem

snsslVD 39980

Virtual deduction proof of snssl 39981. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 ∈ V    ⇒   ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Theorem

snssl 39981

If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4548. The proof of this theorem was automatically generated from snsslVD 39980 using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 ∈ V    ⇒   ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Theorem

snelpwrVD 39982

Virtual deduction proof of snelpwi 5144. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Theorem

unipwrVD 39983

Virtual deduction proof of unipwr 39984. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Theorem

unipwr 39984

A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5150. The proof of this theorem was automatically generated from unipwrVD 39983 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Theorem

sstrALT2VD 39985

Virtual deduction proof of sstrALT2 39986. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Theorem

sstrALT2 39986

Virtual deduction proof of sstr 3828, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 39985 using the command file translate_without_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Theorem

suctrALT2VD 39987

Virtual deduction proof of suctrALT2 39988. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → Tr suc 𝐴)

Theorem

suctrALT2 39988

Virtual deduction proof of suctr 6059. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 39987 using the tools command file translate_without_overwriting_minimize_excluding_duplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Tr 𝐴 → Tr suc 𝐴)

Theorem

elex2VD 39989*

Virtual deduction proof of elex2 3417. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Theorem

elex22VD 39990*

Virtual deduction proof of elex22 3418. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))

Theorem

eqsbc3rVD 39991*

Virtual deduction proof of eqsbc3r 3710. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴))

Theorem

zfregs2VD 39992*

Virtual deduction proof of zfregs2 8906. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ≠ ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))

Theorem

tpid3gVD 39993

Virtual deduction proof of tpid3g 4538. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Theorem

en3lplem1VD 39994*

Virtual deduction proof of en3lplem1 8804. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

Theorem

en3lplem2VD 39995*

Virtual deduction proof of en3lplem2 8805. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

Theorem

en3lpVD 39996

Virtual deduction proof of en3lp 8806. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

20.32.7  Theorems proved using Virtual Deduction with mmj2 assistance

Theorem

simplbi2VD 39997

Virtual deduction proof of simplbi2 496. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
3:1,?: e0a 39923	⊢ ((𝜓 ∧ 𝜒) → 𝜑)
qed:3,?: e0a 39923	⊢ (𝜓 → (𝜒 → 𝜑))

The proof of simplbi2 496 was automatically derived from it. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜑))

Theorem

3ornot23VD 39998

Virtual deduction proof of 3ornot23 39651. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::

⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   (¬ 𝜑 ∧ ¬ 𝜓)   )
2::	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ 𝜑 ∨ 𝜓)   )
3:1,?: e1a 39778	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜑   )
4:1,?: e1a 39778	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜓   )
5:3,4,?: e11 39839	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ (𝜑 ∨ 𝜓)   )
6:2,?: e2 39782	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ (𝜑 ∨ 𝜓))   )
7:5,6,?: e12 39875	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   𝜒   )
8:7:	⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)   )
qed:8:	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Theorem

orbi1rVD 39999

Virtual deduction proof of orbi1r 39652. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜒 ∨ 𝜑)   )
3:2,?: e2 39782	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜑 ∨ 𝜒)   )
4:1,3,?: e12 39875	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜓 ∨ 𝜒)   )
5:4,?: e2 39782	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜒 ∨ 𝜓)   )
6:5:	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓))   )
7::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜒 ∨ 𝜓)   )
8:7,?: e2 39782	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜓 ∨ 𝜒)   )
9:1,8,?: e12 39875	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜑 ∨ 𝜒)   )
10:9,?: e2 39782	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜒 ∨ 𝜑)   )
11:10:	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜓) → (𝜒 ∨ 𝜑))   )
12:6,11,?: e11 39839	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))   )
qed:12:	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

Theorem

bitr3VD 40000

Virtual deduction proof of bitr3 344. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2:1,?: e1a 39778	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜓 ↔ 𝜑)   )
3::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜑 ↔ 𝜒)   )
4:3,?: e2 39782	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜒 ↔ 𝜑)   )
5:2,4,?: e12 39875	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜓 ↔ 𝜒)   )
6:5:	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))   )
qed:6:	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))