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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremee32an 41401 e33an 41375 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 41402 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 41403 e123 41402 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 41404 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 41405 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 41406 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 41407 A virtual deduction elimination rule. The non-virtual deduction form of e000 41407 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 41408 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 41409 Elimination rule identical to mp2an 691. The non-virtual deduction form is the virtual deduction form, which is mp2an 691. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoremeel00cT 41410 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       (⊤ → 𝜒)

TheoremeelTT 41411 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoreme0a 41412 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 41413 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜑𝜓)       𝜓

Theoremeel0cT 41414 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       (⊤ → 𝜓)

TheoremeelT0 41415 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoreme0bi 41416 Elimination rule identical to mpbi 233. The non-virtual deduction form is the virtual deduction form, which is mpbi 233. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

Theoreme0bir 41417 Elimination rule identical to mpbir 234. The non-virtual deduction form is the virtual deduction form, which is mpbir 234. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜑)       𝜓

Theoremuun0.1 41418 Convention notation form of un0.1 41419. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   ((⊤ ∧ 𝜓) → 𝜃)       (𝜓𝜃)

Theoremun0.1 41419 is the constant true, a tautology (see df-tru 1541). 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 41420 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 1541. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)

TheoremuunT1p1 41421 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 41422 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 41423 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 41424 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 41425 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 41426 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 41427 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 672. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremun10 41428 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,      )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )

Theoremun01 41429 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (      ,   𝜑   )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )

Theoremun2122 41430 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 41431 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 41432 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 41433 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 41434 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 41435 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 41436 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 41437 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 41438 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 41439 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 41440 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 41441 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 41442 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 41443 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 41444 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 41445 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 41446 A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1416 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 41447 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 41448 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 41449 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 41450 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 41451 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 41452 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 41453 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 41454 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 41455 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 41456 A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir 1348. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜒𝜓) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)

Theorem3impcombi 41457 A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)
((𝜑𝜓𝜑) → (𝜒𝜃))       ((𝜓𝜑𝜒) → 𝜃)

20.36.6  Theorems proved using Virtual Deduction

TheoremtrsspwALT 41458 Virtual deduction proof of the left-to-right implication of dftr4 5153. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5153 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

TheoremtrsspwALT2 41459 Virtual deduction proof of trsspwALT 41458. This proof is the same as the proof of trsspwALT 41458 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 41460 Short predicate calculus proof of the left-to-right implication of dftr4 5153. 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 41459, which is the virtual deduction proof trsspwALT 41458 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Theoremsspwtr 41461 Virtual deduction proof of the right-to-left implication of dftr4 5153. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 41461 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheoremsspwtrALT 41462 Virtual deduction proof of sspwtr 41461. This proof is the same as the proof of sspwtr 41461 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 41463 Short predicate calculus proof of the right-to-left implication of dftr4 5153. 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 41462, which is the virtual deduction proof sspwtr 41461 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheorempwtrVD 41464 Virtual deduction proof of pwtr 5322; see pwtrrVD 41465 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr 𝒫 𝐴)

TheorempwtrrVD 41465 Virtual deduction proof of pwtr 5322; see pwtrVD 41464 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 41466 The successor of a transitive class is transitive. The proof of https://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 41466 using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/suctrro.html 41466 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 6252 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 41467 Virtual deduction proof of snssiALT 41468. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

TheoremsnssiALT 41468 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4714. This theorem was automatically generated from snssiALTVD 41467 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

TheoremsnsslVD 41469 Virtual deduction proof of snssl 41470. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       ({𝐴} ⊆ 𝐵𝐴𝐵)

Theoremsnssl 41470 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 4692. The proof of this theorem was automatically generated from snsslVD 41469 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 41471 Virtual deduction proof of snelpwi 5314. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

TheoremunipwrVD 41472 Virtual deduction proof of unipwr 41473. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 𝒫 𝐴

Theoremunipwr 41473 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5320. The proof of this theorem was automatically generated from unipwrVD 41472 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 41474 Virtual deduction proof of sstrALT2 41475. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

TheoremsstrALT2 41475 Virtual deduction proof of sstr 3950, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 41474 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.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

TheoremsuctrALT2VD 41476 Virtual deduction proof of suctrALT2 41477. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

TheoremsuctrALT2 41477 Virtual deduction proof of suctr 6252. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 41476 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)

Theoremelex2VD 41478* Virtual deduction proof of elex2 3491. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)

Theoremelex22VD 41479* Virtual deduction proof of elex22 3492. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))

Theoremeqsbc3rVD 41480* Virtual deduction proof of eqsbc3r 3811. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))

Theoremzfregs2VD 41481* Virtual deduction proof of zfregs2 9163. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))

Theoremtpid3gVD 41482 Virtual deduction proof of tpid3g 4682. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Theoremen3lplem1VD 41483* Virtual deduction proof of en3lplem1 9063. (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 41484* Virtual deduction proof of en3lplem2 9064. (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 41485 Virtual deduction proof of en3lp 9065. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance

Theoremsimplbi2VD 41486 Virtual deduction proof of simplbi2 504. 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 41412 ⊢ ((𝜓 ∧ 𝜒) → 𝜑) qed:3,?: e0a 41412 ⊢ (𝜓 → (𝜒 → 𝜑))
The proof of simplbi2 504 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 41487 Virtual deduction proof of 3ornot23 41149. 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 41267 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜑   ) 4:1,?: e1a 41267 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜓   ) 5:3,4,?: e11 41328 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ (𝜑 ∨ 𝜓)   ) 6:2,?: e2 41271 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ (𝜑 ∨ 𝜓))   ) 7:5,6,?: e12 41364 ⊢ (   (¬ 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) → ((𝜒𝜑𝜓) → 𝜒))

Theoremorbi1rVD 41488 Virtual deduction proof of orbi1r 41150. 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 41271 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜑 ∨ 𝜒)   ) 4:1,3,?: e12 41364 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜓 ∨ 𝜒)   ) 5:4,?: e2 41271 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜒 ∨ 𝜓)   ) 6:5: ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓))   ) 7:: ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜒 ∨ 𝜓)   ) 8:7,?: e2 41271 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜓 ∨ 𝜒)   ) 9:1,8,?: e12 41364 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜑 ∨ 𝜒)   ) 10:9,?: e2 41271 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜒 ∨ 𝜑)   ) 11:10: ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜓) → (𝜒 ∨ 𝜑))   ) 12:6,11,?: e11 41328 ⊢ (   (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.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theorembitr3VD 41489 Virtual deduction proof of bitr3 356. 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 41267 ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜓 ↔ 𝜑)   ) 3:: ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜑 ↔ 𝜒)   ) 4:3,?: e2 41271 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜒 ↔ 𝜑)   ) 5:2,4,?: e12 41364 ⊢ (   (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.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Theorem3orbi123VD 41490 Virtual deduction proof of 3orbi123 41151. 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) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ) 2:1,?: e1a 41267 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜑 ↔ 𝜓)   ) 3:1,?: e1a 41267 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜒 ↔ 𝜃)   ) 4:1,?: e1a 41267 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜏 ↔ 𝜂)   ) 5:2,3,?: e11 41328 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃))   ) 6:5,4,?: e11 41328 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))   ) 7:?: ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑 ∨ 𝜒 ∨ 𝜏)) 8:6,7,?: e10 41334 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))   ) 9:?: ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)) 10:8,9,?: e10 41334 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))   ) qed:10: ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂)))

Theoremsbc3orgVD 41491 Virtual deduction proof of the analogue of sbcor 3796 with three disjuncts. 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:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:1,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 3:: ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) 32:3: ⊢ ∀𝑥(((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) 33:1,32,?: e10 41334 ⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒))   ) 4:1,33,?: e11 41328 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒))   ) 5:2,4,?: e11 41328 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 6:1,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))   ) 7:6,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 8:5,7,?: e11 41328 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 9:?: ⊢ ((([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)) 10:8,9,?: e10 41334 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))   ) qed:10: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

Theorem19.21a3con13vVD 41492* Virtual deduction proof of alrim3con13v 41173. 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 → ∀𝑥𝜑)    ▶   (𝜑 → ∀𝑥𝜑)   ) 2:: ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   (𝜓 ∧ 𝜑 ∧ 𝜒)   ) 3:2,?: e2 41271 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   𝜓   ) 4:2,?: e2 41271 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   𝜑   ) 5:2,?: e2 41271 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   𝜒   ) 6:1,4,?: e12 41364 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥𝜑   ) 7:3,?: e2 41271 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥𝜓   ) 8:5,?: e2 41271 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥𝜒   ) 9:7,6,8,?: e222 41276 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)   ) 10:9,?: e2 41271 ⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)   ) 11:10:in2 ⊢ (   (𝜑 → ∀𝑥𝜑)   ▶   ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))   ) qed:11:in1 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) → ((𝜓𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒)))

TheoremexbirVD 41493 Virtual deduction proof of exbir 41118. 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) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ) 2:: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓)   ▶   (𝜑 ∧ 𝜓)   ) 3:: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓), 𝜃   ▶   𝜃   ) 5:1,2,?: e12 41364 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜒 ↔ 𝜃)   ) 6:3,5,?: e32 41398 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃   ▶   𝜒   ) 7:6: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜃 → 𝜒)   ) 8:7: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))   ) 9:8,?: e1a 41267 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃 → 𝜒)))   ) qed:9: ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

TheoremexbiriVD 41494 Virtual deduction proof of exbiri 810. 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 ↔ 𝜃)) 2:: ⊢ (   𝜑   ▶   𝜑   ) 3:: ⊢ (   𝜑   ,   𝜓   ▶   𝜓   ) 4:: ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   ) 5:2,1,?: e10 41334 ⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   ) 6:3,5,?: e21 41370 ⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   ) 7:4,6,?: e32 41398 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   ) 8:7: ⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   ) 9:8: ⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   ) qed:9: ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremrspsbc2VD 41495* Virtual deduction proof of rspsbc2 41174. 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:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:: ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   𝐶 ∈ 𝐷   ) 3:: ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑   ) 4:1,3,?: e13 41388 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑   ) 5:1,4,?: e13 41388 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑   ) 6:2,5,?: e23 41395 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   ) 7:6: ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   ) 8:7: ⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   ) qed:8: ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Theorem3impexpVD 41496 Virtual deduction proof of 3impexp 1355. 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 ∧ 𝜒) → 𝜃)   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ) 2:: ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) 3:1,2,?: e10 41334 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)   ) 4:3,?: e1a 41267 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))   ) 5:4,?: e1a 41267 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ) 6:5: ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒 → 𝜃)))) 7:: ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ) 8:7,?: e1a 41267 ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))   ) 9:8,?: e1a 41267 ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)   ) 10:2,9,?: e01 41331 ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ) 11:10: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)) qed:6,11,?: e00 41408 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Theorem3impexpbicomVD 41497 Virtual deduction proof of 3impexpbicom 41119. 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 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ) 2:: ⊢ ((𝜃 ↔ 𝜏) ↔ (𝜏 ↔ 𝜃)) 3:1,2,?: e10 41334 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃))   ) 4:3,?: e1a 41267 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ▶   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ) 5:4: ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))) 6:: ⊢ (   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ▶   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ) 7:6,?: e1a 41267 ⊢ (   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃))   ) 8:7,2,?: e10 41334 ⊢ (   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ) 9:8: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))) qed:5,9,?: e00 41408 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theorem3impexpbicomiVD 41498 Virtual deduction proof of 3impexpbicomi 41120. 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) → (𝜃 ↔ 𝜏)) qed:1,?: e0a 41412 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))

TheoremsbcoreleleqVD 41499* Virtual deduction proof of sbcoreleleq 41175. 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:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:1,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)   ) 3:1,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)   ) 4:1,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝐴)   ) 5:2,3,4,?: e111 41314 ⊢ (   𝐴 ∈ 𝐵   ▶   ((𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦))   ) 6:1,?: e1a 41267 ⊢ (   𝐴 ∈ 𝐵    ▶   ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦))   ) 7:5,6: e11 41328 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))   ) qed:7: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))

Theoremhbra2VD 41500* Virtual deduction proof of nfra2 3217. 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:: ⊢ (∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑) 2:: ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑) 3:1,2,?: e00 41408 ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑) 4:2: ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑) 5:4,?: e0a 41412 ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑) qed:3,5,?: e00 41408 ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝑥𝐴𝑦𝐵 𝜑)

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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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