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Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreme13 38801 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )    &   (𝜓 → (𝜏𝜂))       (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Theoreme13an 38802 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   ▶   𝜏   )    &   ((𝜓𝜏) → 𝜂)       (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Theoremee13an 38803 e13an 38802 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 38804 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 38805 e31 38804 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 38806 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 38807 e31an 38806 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 38808 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 38809 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 38810 e23an 38809 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 38811 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 38812 e32 38811 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 38813 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 38814 e33an 38788 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 38815 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 38816 e123 38815 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 38817 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 38818 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 38819 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 38820 A virtual deduction elimination rule. The non-virtual deduction form of e000 38820 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 38821 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 38822 Elimination rule identical to mp2an 708. The non-virtual deduction form is the virtual deduction form, which is mp2an 708. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoremeel00cT 38823 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       (⊤ → 𝜒)

TheoremeelTT 38824 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoreme0a 38825 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 38826 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜑𝜓)       𝜓

Theoremeel0cT 38827 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       (⊤ → 𝜓)

TheoremeelT0 38828 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoreme0bi 38829 Elimination rule identical to mpbi 220. The non-virtual deduction form is the virtual deduction form, which is mpbi 220. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

Theoreme0bir 38830 Elimination rule identical to mpbir 221. The non-virtual deduction form is the virtual deduction form, which is mpbir 221. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜑)       𝜓

Theoremuun0.1 38831 Convention notation form of un0.1 38832. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   ((⊤ ∧ 𝜓) → 𝜃)       (𝜓𝜃)

Theoremun0.1 38832 is the constant true, a tautology (see df-tru 1485). 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 38833 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accomodate a possible future version of df-tru 1485. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)

TheoremuunT1p1 38834 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 38835 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 38836 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 38837 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 38838 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 38839 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 38840 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 862. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremun10 38841 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,      )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )

Theoremun01 38842 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (      ,   𝜑   )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )

Theoremun2122 38843 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 38844 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 38845 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 38846 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 38847 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 38848 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 38849 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 38850 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 38851 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 38852 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 38853 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 38854 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 38855 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 38856 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 38857 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 38858 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 38859 A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1382 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 38860 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 38861 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 38862 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 38863 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 38864 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 38865 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 38866 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 38867 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 38868 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 38869 A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir 1381. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜒𝜓) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)

Theorem3impcombi 38870 A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)
((𝜑𝜓𝜑) → (𝜒𝜃))       ((𝜓𝜑𝜒) → 𝜃)

20.31.6  Theorems proved using Virtual Deduction

TheoremtrsspwALT 38871 Virtual deduction proof of the left-to-right implication of dftr4 4755. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4755 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

TheoremtrsspwALT2 38872 Virtual deduction proof of trsspwALT 38871. This proof is the same as the proof of trsspwALT 38871 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 38873 Short predicate calculus proof of the left-to-right implication of dftr4 4755. 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 38872, which is the virtual deduction proof trsspwALT 38871 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Theoremsspwtr 38874 Virtual deduction proof of the right-to-left implication of dftr4 4755. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 38874 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheoremsspwtrALT 38875 Virtual deduction proof of sspwtr 38874. This proof is the same as the proof of sspwtr 38874 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)

TheoremcsbabgOLD 38876* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 4006 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})

TheoremcsbunigOLD 38877 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 22-Aug-2018. Use csbuni 4464 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Theoremcsbfv12gALTOLD 38878 Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 38961. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 6229 instead. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

TheoremcsbxpgOLD 38879 Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5594 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐷𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))

TheoremcsbingOLD 38880 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 4008 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

TheoremcsbresgOLD 38881 Distribute proper substitution through the restriction of a class. csbresgOLD 38881 is derived from the virtual deduction proof csbresgVD 38957. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5397 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

TheoremcsbrngOLD 38882 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5594 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)

Theoremcsbima12gALTOLD 38883 Move class substitution in and out of the image of a function. The proof is derived from the virtual deduction proof csbima12gALTVD 38959. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 6229 instead. (Proof modification is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

TheoremsspwtrALT2 38884 Short predicate calculus proof of the right-to-left implication of dftr4 4755. 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 38875, which is the virtual deduction proof sspwtr 38874 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheorempwtrVD 38885 Virtual deduction proof of pwtr 4919; see pwtrrVD 38886 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr 𝒫 𝐴)

TheorempwtrrVD 38886 Virtual deduction proof of pwtr 4919; see pwtrVD 38885 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 38887 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 38887 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 5806 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 38888 Virtual deduction proof of snssiALT 38889. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

TheoremsnssiALT 38889 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4337. This theorem was automatically generated from snssiALTVD 38888 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

TheoremsnsslVD 38890 Virtual deduction proof of snssl 38891. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       ({𝐴} ⊆ 𝐵𝐴𝐵)

Theoremsnssl 38891 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 4314. The proof of this theorem was automatically generated from snsslVD 38890 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 38892 Virtual deduction proof of snelpwi 4910. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

TheoremunipwrVD 38893 Virtual deduction proof of unipwr 38894. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 𝒫 𝐴

Theoremunipwr 38894 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4916. The proof of this theorem was automatically generated from unipwrVD 38893 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 38895 Virtual deduction proof of sstrALT2 38896. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

TheoremsstrALT2 38896 Virtual deduction proof of sstr 3609, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 38895 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 38897 Virtual deduction proof of suctrALT2 38898. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

TheoremsuctrALT2 38898 Virtual deduction proof of suctr 5806. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 38897 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 38899* Virtual deduction proof of elex2 3214. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)

Theoremelex22VD 38900* Virtual deduction proof of elex22 3215. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))

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