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Theorem List for Metamath Proof Explorer - 39901-40000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremee13an 39901 e13an 39900 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒 → (𝜃𝜏)))    &   ((𝜓𝜏) → 𝜂)       (𝜑 → (𝜒 → (𝜃𝜂)))
 
Theoreme31 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee31 39903 e31 39902 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑𝜏)    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme31an 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   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee31an 39905 e31an 39904 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑𝜏)    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme23 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
 
Theoreme23an 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   )    &   ((𝜒𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
 
Theoremee23an 39908 e23an 39907 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   ((𝜒𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜃𝜂)))
 
Theoreme32 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee32 39910 e32 39909 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme32an 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   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee32an 39912 e33an 39886 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme123 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 → (𝜂𝜁)))       (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )
 
Theoremee123 39914 e123 39913 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒 → (𝜏𝜂)))    &   (𝜓 → (𝜃 → (𝜂𝜁)))       (𝜑 → (𝜒 → (𝜏𝜁)))
 
Theoremel123 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   )    &   ((𝜓𝜃𝜂) → 𝜁)       (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )
 
Theoreme233 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 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )
 
Theoreme323 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 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )
 
Theoreme000 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 → (𝜒𝜃)))       𝜃
 
Theoreme00 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 → (𝜓𝜒))       𝜒
 
Theoreme00an 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    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremeel00cT 39921 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       (⊤ → 𝜒)
 
TheoremeelTT 39922 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoreme0a 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    &   (𝜑𝜓)       𝜓
 
TheoremeelT 39924 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜑𝜓)       𝜓
 
Theoremeel0cT 39925 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       (⊤ → 𝜓)
 
TheoremeelT0 39926 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoreme0bi 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    &   (𝜑𝜓)       𝜓
 
Theoreme0bir 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    &   (𝜓𝜑)       𝜓
 
Theoremuun0.1 39929 Convention notation form of un0.1 39930. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   ((⊤ ∧ 𝜓) → 𝜃)       (𝜓𝜃)
 
Theoremun0.1 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   )
 
TheoremuunT1 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)       (𝜑𝜓)
 
TheoremuunT1p1 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)       (𝜑𝜓)
 
TheoremuunT21 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.)
((⊤ ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun121 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 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun121p1 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.)
(((𝜑𝜓) ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun132 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 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun132p1 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.)
(((𝜓𝜒) ∧ 𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremanabss7p1 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.)
(((𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremun10 39939 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,      )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )
 
Theoremun01 39940 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (      ,   𝜑   )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )
 
Theoremun2122 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.)
(((𝜑𝜓) ∧ 𝜓𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun2131 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.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun2131p1 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.)
(((𝜑𝜒) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
TheoremuunTT1 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)       (𝜑𝜓)
 
TheoremuunTT1p1 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)       (𝜑𝜓)
 
TheoremuunTT1p2 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)       (𝜑𝜓)
 
TheoremuunT11 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.)
((⊤ ∧ 𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
TheoremuunT11p1 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)       (𝜑𝜓)
 
TheoremuunT11p2 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.)
((𝜑𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)
 
TheoremuunT12 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.)
((⊤ ∧ 𝜑𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p1 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.)
((⊤ ∧ 𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p2 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)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p3 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)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p4 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.)
((𝜑𝜓 ∧ ⊤) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p5 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.)
((𝜓𝜑 ∧ ⊤) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun111 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.)
((𝜑𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
Theorem3anidm12p1 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.)
((𝜑𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theorem3anidm12p2 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.)
((𝜓𝜑𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun123 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.)
((𝜑𝜒𝜓) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p1 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.)
((𝜓𝜑𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p2 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.)
((𝜒𝜑𝜓) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p3 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.)
((𝜓𝜒𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p4 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.)
((𝜒𝜓𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun2221 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.)
((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremuun2221p1 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 ∧ (𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremuun2221p2 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.)
(((𝜓𝜑) ∧ 𝜑𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theorem3impdirp1 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.)
(((𝜒𝜓) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)
 
Theorem3impcombi 39968 A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)
((𝜑𝜓𝜑) → (𝜒𝜃))       ((𝜓𝜑𝜒) → 𝜃)
 
20.32.6  Theorems proved using Virtual Deduction
 
TheoremtrsspwALT 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 𝐴𝐴 ⊆ 𝒫 𝐴)
 
TheoremtrsspwALT2 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 𝐴𝐴 ⊆ 𝒫 𝐴)
 
TheoremtrsspwALT3 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 𝐴𝐴 ⊆ 𝒫 𝐴)
 
Theoremsspwtr 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)
 
TheoremsspwtrALT 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)
 
TheoremsspwtrALT2 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)
 
TheorempwtrVD 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)
 
TheorempwtrrVD 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)
 
TheoremsuctrALT 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)
 
TheoremsnssiALTVD 39978 Virtual deduction proof of snssiALT 39979. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)
 
TheoremsnssiALT 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.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)
 
TheoremsnsslVD 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} ⊆ 𝐵𝐴𝐵)
 
Theoremsnssl 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} ⊆ 𝐵𝐴𝐵)
 
TheoremsnelpwrVD 39982 Virtual deduction proof of snelpwi 5144. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
 
TheoremunipwrVD 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
 
Theoremunipwr 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
 
TheoremsstrALT2VD 39985 Virtual deduction proof of sstrALT2 39986. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
TheoremsstrALT2 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 translatewithout_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.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
TheoremsuctrALT2VD 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)
 
TheoremsuctrALT2 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 translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)
 
Theoremelex2VD 39989* Virtual deduction proof of elex2 3417. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)
 
Theoremelex22VD 39990* Virtual deduction proof of elex22 3418. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 
Theoremeqsbc3rVD 39991* Virtual deduction proof of eqsbc3r 3710. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
 
Theoremzfregs2VD 39992* Virtual deduction proof of zfregs2 8906. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))
 
Theoremtpid3gVD 39993 Virtual deduction proof of tpid3g 4538. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremen3lplem1VD 39994* Virtual deduction proof of en3lplem1 8804. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 
Theoremen3lplem2VD 39995* Virtual deduction proof of en3lplem2 8805. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 
Theoremen3lpVD 39996 Virtual deduction proof of en3lp 8806. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
 
20.32.7  Theorems proved using Virtual Deduction with mmj2 assistance
 
Theoremsimplbi2VD 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 ↔ (𝜓𝜒))
3:1,?: e0a 39923 ((𝜓𝜒) → 𝜑)
qed:3,?: e0a 39923 (𝜓 → (𝜒𝜑))
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 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theorem3ornot23VD 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)   ,   (𝜒𝜑 𝜓)   ▶   (𝜒𝜑𝜓)   )
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)   ,   (𝜒𝜑 𝜓)   ▶   (𝜒 ∨ (𝜑𝜓))   )
7:5,6,?: e12 39875 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑 𝜓)   ▶   𝜒   )
8:7: (   𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒 𝜑𝜓) → 𝜒)   )
qed:8: ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 𝜑𝜓) → 𝜒))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
 
Theoremorbi1rVD 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:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜒𝜑)   )
3:2,?: e2 39782 (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜑𝜒)   )
4:1,3,?: e12 39875 (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜓𝜒)   )
5:4,?: e2 39782 (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜒𝜓)   )
6:5: (   (𝜑𝜓)   ▶   ((𝜒𝜑) → (𝜒𝜓))   )
7:: (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜒𝜓)   )
8:7,?: e2 39782 (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜓𝜒)   )
9:1,8,?: e12 39875 (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜑𝜒)   )
10:9,?: e2 39782 (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜒𝜑)   )
11:10: (   (𝜑𝜓)   ▶   ((𝜒𝜓) → (𝜒𝜑))   )
12:6,11,?: e11 39839 (   (𝜑𝜓)   ▶   ((𝜒 𝜑) ↔ (𝜒𝜓))   )
qed:12: ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theorembitr3VD 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:: (   (𝜑𝜓)   ▶   (𝜑 𝜓)   )
2:1,?: e1a 39778 (   (𝜑𝜓)   ▶   (𝜓 𝜑)   )
3:: (   (𝜑𝜓)   ,   (𝜑𝜒)    ▶   (𝜑𝜒)   )
4:3,?: e2 39782 (   (𝜑𝜓)   ,   (𝜑𝜒)    ▶   (𝜒𝜑)   )
5:2,4,?: e12 39875 (   (𝜑𝜓)   ,   (𝜑𝜒)    ▶   (𝜓𝜒)   )
6:5: (   (𝜑𝜓)   ▶   ((𝜑 𝜒) → (𝜓𝜒))   )
qed:6: ((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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