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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembi123impib 40701 3impib 1108 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ ((𝜓𝜒) ↔ 𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi13impia 40702 3impia 1109 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑𝜓) ↔ (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi123impia 40703 3impia 1109 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑𝜓) ↔ (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi33imp12 40704 3imp 1103 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi23imp13 40705 3imp 1103 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi13imp23 40706 3imp 1103 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi13imp2 40707 Similar to 3imp 1103 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi12imp3 40708 Similar to 3imp 1103 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi23imp1 40709 Similar to 3imp 1103 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembi123imp0 40710 Similar to 3imp 1103 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem4animp1 40711 A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)
((𝜑𝜓𝜒) → (𝜏𝜃))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem4an31 40712 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((𝜒𝜓) ∧ 𝜑) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem4an4132 40713 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((𝜃𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremexpcomdg 40714 Biconditional form of expcomd 417. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜒 → (𝜓𝜃))))
 
20.35.3  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 40715 idn3 40829 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜒)))
 
Theoremee222 40716 e222 40850 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))
 
Theoremee3bir 40717 Right-biconditional form of e3 40951 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜏𝜃)       (𝜑 → (𝜓 → (𝜒𝜏)))
 
Theoremee13 40718 e13 40962 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒 → (𝜃𝜏)))    &   (𝜓 → (𝜏𝜂))       (𝜑 → (𝜒 → (𝜃𝜂)))
 
Theoremee121 40719 e121 40870 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑𝜏)    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))
 
Theoremee122 40720 e122 40867 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒𝜏))    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))
 
Theoremee333 40721 e333 40947 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜒𝜁)))
 
Theoremee323 40722 e323 40980 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜒𝜁)))
 
Theorem3ornot23 40723 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 41061. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
 
Theoremorbi1r 40724 orbi1 911 with order of disjuncts reversed. Derived from orbi1rVD 41062. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theorem3orbi123 40725 pm4.39 970 with a 3-conjunct antecedent. This proof is 3orbi123VD 41064 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂)))
 
Theoremsyl5imp 40726 Closed form of syl5 34. Derived automatically from syl5impVD 41077. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜓) → (𝜑 → (𝜃𝜒))))
 
Theoremimpexpd 40727 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed, it was minimized. The completed User's Proof before minimization is not shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (((𝜓𝜒) → 𝜃) ↔ (𝜓 → (𝜒 𝜃)))
qed:1: ((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
 
Theoremcom3rgbi 40728 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: ((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))
2:: ((𝜑 → (𝜒 → (𝜓𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))
3:1,2: ((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))
4:: ((𝜒 → (𝜑 → (𝜓𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))
5:: ((𝜑 → (𝜒 → (𝜓𝜃))) → (𝜑 → (𝜓 → (𝜒𝜃))))
6:4,5: ((𝜒 → (𝜑 → (𝜓𝜃))) → (𝜑 → (𝜓 → (𝜒𝜃))))
qed:3,6: ((𝜑 → (𝜓 → (𝜒𝜃))) ↔ (𝜒 → (𝜑 → (𝜓𝜃))))
((𝜑 → (𝜓 → (𝜒𝜃))) ↔ (𝜒 → (𝜑 → (𝜓𝜃))))
 
Theoremimpexpdcom 40729 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: ((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
2:: ((𝜓 → (𝜒 → (𝜑𝜃))) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
qed:1,2: ((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
 
Theoremee1111 40730 Non-virtual deduction form of e1111 40889. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1:: (𝜑𝜓)
h2:: (𝜑𝜒)
h3:: (𝜑𝜃)
h4:: (𝜑𝜏)
h5:: (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))
6:1,5: (𝜑 → (𝜒 → (𝜃 → (𝜏𝜂))))
7:6: (𝜒 → (𝜑 → (𝜃 → (𝜏𝜂))))
8:2,7: (𝜑 → (𝜑 → (𝜃 → (𝜏𝜂))))
9:8: (𝜑 → (𝜃 → (𝜏𝜂)))
10:9: (𝜃 → (𝜑 → (𝜏𝜂)))
11:3,10: (𝜑 → (𝜑 → (𝜏𝜂)))
12:11: (𝜑 → (𝜏𝜂))
13:12: (𝜏 → (𝜑𝜂))
14:4,13: (𝜑 → (𝜑𝜂))
qed:14: (𝜑𝜂)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))       (𝜑𝜂)
 
Theorempm2.43bgbi 40731 Logical equivalence of a 2-left-nested implication and a 1-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1:: ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜑 → (𝜑 → (𝜓𝜒))))
2:: ((𝜑 → (𝜑 → (𝜓𝜒))) → (𝜑 → (𝜓𝜒)))
3:1,2: ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜑 → (𝜓𝜒)))
4:: ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
5:3,4: ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))
6:: ((𝜓 → (𝜑𝜒)) → (𝜑 → (𝜓 → (𝜑𝜒))))
qed:5,6: ((𝜑 → (𝜓 → (𝜑𝜒))) ↔ (𝜓 → (𝜑𝜒)))
((𝜑 → (𝜓 → (𝜑𝜒))) ↔ (𝜓 → (𝜑𝜒)))
 
Theorempm2.43cbi 40732 Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1:: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) → (𝜑 → (𝜓 → (𝜑 → (𝜒𝜃)))))
2:: ((𝜑 → (𝜓 → (𝜑 → (𝜒𝜃))) ) → (𝜓 → (𝜑 → (𝜒𝜃))))
3:1,2: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) → (𝜓 → (𝜑 → (𝜒𝜃))))
4:: ((𝜓 → (𝜑 → (𝜒𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))
5:3,4: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) → (𝜓 → (𝜒 → (𝜑𝜃))))
6:: ((𝜓 → (𝜒 → (𝜑𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))))
qed:5,6: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
 
Theoremee233 40733 Non-virtual deduction form of e233 40979. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1:: (𝜑 → (𝜓𝜒))
h2:: (𝜑 → (𝜓 → (𝜃𝜏)))
h3:: (𝜑 → (𝜓 → (𝜃𝜂)))
h4:: (𝜒 → (𝜏 → (𝜂𝜁)))
5:1,4: (𝜑 → (𝜓 → (𝜏 → (𝜂𝜁))) )
6:5: (𝜏 → (𝜑 → (𝜓 → (𝜂𝜁))) )
7:2,6: (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
8:7: (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 𝜁)))))
9:8: (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))) )
10:9: (𝜑 → (𝜓 → (𝜃 → (𝜂𝜁))) )
11:10: (𝜂 → (𝜑 → (𝜓 → (𝜃𝜁))) )
12:3,11: (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
13:12: (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 𝜁)))))
14:13: (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))) )
qed:14: (𝜑 → (𝜓 → (𝜃𝜁)))
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜑 → (𝜓 → (𝜃𝜂)))    &   (𝜒 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜃𝜁)))
 
Theoremimbi13 40734 Join three logical equivalences to form equivalence of implications. imbi13 40734 is imbi13VD 41088 without virtual deductions and was automatically derived from imbi13VD 41088 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))
 
Theoremee33 40735 Non-virtual deduction form of e33 40948. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1:: (𝜑 → (𝜓 → (𝜒𝜃)))
h2:: (𝜑 → (𝜓 → (𝜒𝜏)))
h3:: (𝜃 → (𝜏𝜂))
4:1,3: (𝜑 → (𝜓 → (𝜒 → (𝜏𝜂))))
5:4: (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂))))
6:2,5: (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 (𝜒𝜂))))))
7:6: (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 𝜂)))))
8:7: (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
qed:8: (𝜑 → (𝜓 → (𝜒𝜂)))
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoremcon5 40736 Biconditional contraposition variation. This proof is con5VD 41114 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
 
Theoremcon5i 40737 Inference form of con5 40736. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ 𝜓)       𝜑𝜓)
 
Theoremexlimexi 40738 Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (∃𝑥𝜑 → (𝜑𝜓))       (∃𝑥𝜑𝜓)
 
Theoremsb5ALT 40739* Equivalence for substitution. Alternate proof of sb5 2268. This proof is sb5ALTVD 41127 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremeexinst01 40740 exinst01 40839 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝜑 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (𝜑𝜒)
 
Theoremeexinst11 40741 exinst11 40840 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (𝜑𝜒)
 
Theoremvk15.4j 40742 Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 40742 is vk15.4jVD 41128 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))    &   (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))    &    ¬ ∀𝑥(𝜏𝜑)       (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)
 
TheoremnotnotrALT 40743 Converse of double negation. Alternate proof of notnotr 132. This proof is notnotrALTVD 41129 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)
 
Theoremcon3ALT2 40744 Contraposition. Alternate proof of con3 156. This proof is con3ALTVD 41130 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Theoremssralv2 40745* Quantification restricted to a subclass for two quantifiers. ssralv 4032 for two quantifiers. The proof of ssralv2 40745 was automatically generated by minimizing the automatically translated proof of ssralv2VD 41080. The automatic translation is by the tools program translate_without_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
 
Theoremsbc3or 40746 sbcor 3821 with a 3-disjuncts. This proof is sbc3orgVD 41065 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremalrim3con13v 40747* Closed form of alrimi 2204 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 41066 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) → ((𝜓𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒)))
 
Theoremrspsbc2 40748* rspsbc 3861 with two quantifying variables. This proof is rspsbc2VD 41069 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
 
Theoremsbcoreleleq 40749* Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 41073. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
 
Theoremtratrb 40750* If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 41075. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
 
TheoremordelordALT 40751 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6207 using the Axiom of Regularity indirectly through dford2 9072. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that E Fr 𝐴 because this is inferred by the Axiom of Regularity. ordelordALT 40751 is ordelordALTVD 41081 without virtual deductions and was automatically derived from ordelordALTVD 41081 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((Ord 𝐴𝐵𝐴) → Ord 𝐵)
 
Theoremsbcim2g 40752 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3819. sbcim2g 40752 is sbcim2gVD 41089 without virtual deductions and was automatically derived from sbcim2gVD 41089 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
 
Theoremsbcbi 40753 Implication form of sbcbii 3828. sbcbi 40753 is sbcbiVD 41090 without virtual deductions and was automatically derived from sbcbiVD 41090 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremtrsbc 40754* Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 40754 is trsbcVD 41091 without virtual deductions and was automatically derived from trsbcVD 41091 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
 
TheoremtruniALT 40755* The union of a class of transitive sets is transitive. Alternate proof of truni 5178. truniALT 40755 is truniALTVD 41092 without virtual deductions and was automatically derived from truniALTVD 41092 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
TheoremonfrALTlem5 40756* Lemma for onfrALT 40763. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
 
TheoremonfrALTlem4 40757* Lemma for onfrALT 40763. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
 
TheoremonfrALTlem3 40758* Lemma for onfrALT 40763. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
 
Theoremggen31 40759* gen31 40835 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))
 
TheoremonfrALTlem2 40760* Lemma for onfrALT 40763. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
 
Theoremcbvexsv 40761* A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
 
TheoremonfrALTlem1 40762* Lemma for onfrALT 40763. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
 
TheoremonfrALT 40763 The membership relation is foundational on the class of ordinal numbers. onfrALT 40763 is an alternate proof of onfr 6224. onfrALTVD 41105 is the Virtual Deduction proof from which onfrALT 40763 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6224 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 41105. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
E Fr On
 
Theorem19.41rg 40764 Closed form of right-to-left implication of 19.41 2228, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 41116. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
 
Theoremopelopab4 40765* Ordered pair membership in a class abstraction of pairs. Compare to elopab 5406. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
 
Theorem2pm13.193 40766 pm13.193 40623 for two variables. pm13.193 40623 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 41117. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
 
Theoremhbntal 40767 A closed form of hbn 2295. hbnt 2294 is another closed form of hbn 2295. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
 
Theoremhbimpg 40768 A closed form of hbim 2299. Derived from hbimpgVD 41118. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑𝜓) → ∀𝑥(𝜑𝜓)))
 
Theoremhbalg 40769 Closed form of hbal 2164. Derived from hbalgVD 41119. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theoremhbexg 40770 Closed form of nfex 2335. Derived from hbexgVD 41120. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theoremax6e2eq 40771* Alternate form of ax6e 2394 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 40771 is derived from ax6e2eqVD 41121. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
 
Theoremax6e2nd 40772* If at least two sets exist (dtru 5263) , then the same is true expressed in an alternate form similar to the form of ax6e 2394. ax6e2nd 40772 is derived from ax6e2ndVD 41122. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 
Theoremax6e2ndeq 40773* "At least two sets exist" expressed in the form of dtru 5263 is logically equivalent to the same expressed in a form similar to ax6e 2394 if dtru 5263 is false implies 𝑢 = 𝑣. ax6e2ndeq 40773 is derived from ax6e2ndeqVD 41123. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 
Theorem2sb5nd 40774* Equivalence for double substitution 2sb5 2274 without distinct 𝑥, 𝑦 requirement. 2sb5nd 40774 is derived from 2sb5ndVD 41124. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
 
Theorem2uasbanh 40775* Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 40775 is derived from 2uasbanhVD 41125. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))       (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
 
Theorem2uasban 40776* Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
 
Theoreme2ebind 40777 Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 40777 is derived from e2ebindVD 41126. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
 
Theoremelpwgded 40778 elpwgdedVD 41131 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ V)    &   (𝜓𝐴𝐵)       ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)
 
Theoremtrelded 40779 Deduction form of trel 5171. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Tr 𝐴)    &   (𝜓𝐵𝐶)    &   (𝜒𝐶𝐴)       ((𝜑𝜓𝜒) → 𝐵𝐴)
 
Theoremjaoded 40780 Deduction form of jao 954. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))    &   (𝜂 → (𝜓𝜏))       ((𝜑𝜃𝜂) → 𝜒)
 
TheoremsbtT 40781 A substitution into a theorem remains true. sbt 2062 with the existence of no virtual hypotheses for the hypothesis expressed as the empty virtual hypothesis collection. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)       [𝑦 / 𝑥]𝜑
 
Theoremnot12an2impnot1 40782 If a double conjunction is false and the second conjunct is true, then the first conjunct is false. https://us.metamath.org/other/completeusersproof/not12an2impnot1vd.html is the Virtual Deduction proof verified by automatically transforming it into the Metamath proof of not12an2impnot1 40782 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 40782 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.)
((¬ (𝜑𝜓) ∧ 𝜓) → ¬ 𝜑)
 
20.35.4  What is Virtual Deduction?
 
Syntaxwvd1 40783 A Virtual Deduction proof in a Hilbert-style deductive system is the analogue of a sequent calculus proof. A theorem is proven in a Gentzen system in order to prove more directly, which may be more intuitive and easier for some people. The analogue of this proof in Metamath's Hilbert-style system is verified by the Metamath program.

Natural Deduction is a well-known proof method orignally proposed by Gentzen in 1935 and comprehensively summarized by Prawitz in his 1965 monograph "Natural deduction: a proof-theoretical study". Gentzen wished to construct "a formalism that comes as close as possible to natural reasoning". Natural deduction is a response to dissatisfaction with axiomatic proofs such as Hilbert-style axiomatic proofs, which the proofs of Metamath are. In 1926, in Poland, Lukasiewicz advocated a more natural treatment of logic. Jaskowski made the earliest attempts at defining a more natural deduction. Natural deduction in its modern form was independently proposed by Gentzen.

Sequent calculus, the chief alternative to Natural Deduction, was created by Gentzen. The following is an excerpt from Stephen Cole Kleene's seminal 1952 book "Introduction to Metamathematics", which contains the first formulation of sequent calculus in the modern style. Kleene states on page 440:

. . . the proof (of Gentzen's Hauptsatz) breaks down into a list of cases, each of which is simple to handle. . . . Gentzen's normal form for proofs in the predicate calculus requires a different classification of the deductive steps than is given by the postulates of the formal system of predicate calculus of Chapter IV (Section 19). The implication symbol (the Metamath symbol for implication has been substituted here for the symbol used by Kleene) has to be separated in its role of mediating inferences from its role as a component symbol of the formula being proved. In the former role it will be replaced by a new formal symbol (read "gives" or "entails"), to which properties will be assigned similar to those of the informal symbol in our former derived rules.

Gentzen's classification of the deductive operations is made explicit by setting up a new formal system of the predicate calculus. The formal system of propositional and predicate calculus studied previously (Chapters IV ff.) we call now a "Hilbert-type system", and denote by H. Precisely, H denotes any one or a particular one of several systems, according to whether we are considering propositional calculus or predicate calculus, in the classical or the intuitionistic version (Section 23), and according to the sense in which we are using "term" and "formula" (Sections 117,25,31,37,72-76). The same respective choices will apply to the "Gentzen-type system G1" which we introduce now and the G2, G3 and G3a later.

The transformation or deductive rules of G1 will apply to objects which are not formulas of the system H, but are built from them by an additional formation rule, so we use a new term "sequent" for these objects. (Gentzen says "Sequenz", which we translate as "sequent", because we have already used "sequence" for any succession of objects, where the German is "Folge".) A sequent is a formal expression of the form 𝜑, . . . , 𝜓 𝜒, . . . , 𝜃 where 𝜑 , . . . , 𝜓 and 𝜒, . . . , 𝜃 are seqences of a finite number of 0 or more formulas (substituting Metamath notation for Kleene's notation). The part 𝜑, . . . , 𝜓 is the antecedent, and 𝜒, . . . , 𝜃 the succedent of the sequent 𝜑, . . . , 𝜓 𝜒, . . . , 𝜃.

When the antecedent and the succedent each have a finite number of 1 or more formulas, the sequent 𝜑, . . . , 𝜓 𝜒, . . . 𝜃 has the same interpretation for G1 as the formula ((𝜑. . . 𝜓) → (𝜒. . . 𝜃)) for H. The interpretation extends to the case of an antecedent of 0 formulas by regarding (𝜑. . . 𝜓) for 0 formulas (the "empty conjunction") as true and (𝜒. . . 𝜃) for 0 formulas (the "empty disjunction") as false.

. . . As in Chapter V, we use Greek capitals . . . to stand for finite sequences of zero or more formulas, but now also as antecedent (succedent), or parts of antecedent (succedent), with separating formal commas included. . . . (End of Kleene excerpt)

In chapter V entitled "Formal Deduction" Kleene states, on page 86:

Section 20. Formal deduction. Formal proofs of even quite elementary theorems tend to be long. As a price for having analyzed logical deduction into simple steps, more of those steps have to be used.

The purpose of formalizing a theory is to get an explicit definition of what constitutes proof in the theory. Having achieved this, there is no need always to appeal directly to the definition. The labor required to establish the formal provability of formulas can be greatly lessened by using metamathematical theorems concerning the existence of formal proofs. If the demonstrations of those theorems do have the finitary character which metamathematics is supposed to have, the demonstrations will indicate, at least implicitly, methods for obtaining the formal proofs. The use of the metamathematical theorems then amounts to abbreviation, often of very great extent, in the presentation of formal proofs.

The simpler of such metamathematical theorems we shall call derived rules, since they express principles which can be said to be derived from the postulated rules by showing that the use of them as additional methods of inference does not increase the class of provable formulas. We shall seek by means of derived rules to bring the methods for establishing the facts of formal provability as close as possible to the informal methods of the theory which is being formalized.

In setting up the formal system, proof was given the simplest possible structure, consisting of a single sequence of formulas. Some of our derived rules, called "direct rules", will serve to abbreviate for us whole segments of such a sequence; we can then, so to speak, use these segments as prefabricated units in building proofs.

But also, in mathematical practice, proofs are common which have a more complicated structure, employing "subsidiary deduction", i.e., deduction under assumptions for the sake of the argument, which assumptions are subsequently discharged. For example, subsidiary deduction is used in a proof by reductio ad absurdum, and less obtrusively when we place the hypothesis of a theorem on a par with proved propositions to deduce the conclusion. Other derived rules, called "subsidiary deduction rules", will give us this kind of procedure.

We now introduce, by a metamathematical definition, the notion of "formal deducibility under assumptions". Given a list 𝜑, . . . 𝜓 of 0 or more (occurrences of) formulas, a finite sequence of one or more (occurrences of) formulas is called a (formal) deduction from the assumption formulas 𝜑, . . . 𝜓, if each formula of the sequence is either one of the formulas 𝜑, . . . 𝜓, or an axiom, or an immediate consequence of preceding formulas of a sequence. A deduction is said to be deducible from the assumption formulas (in symbols, 𝜑,. . . . ,. 𝜓𝜒), and is called the conclusion (or endformula) of the deduction. (The symbol may be read "yields".) (End of Kleene excerpt)

Gentzen's normal form is a certain direct fashion for proofs and deductions. His sequent calculus, formulated in the modern style by Kleene, is the classical system G1. In this system, the new formal symbol has properties similar to the informal symbol of Kleene's above language of formal deducibility under assumptions.

Kleene states on page 440:

. . . This leads us to inquire whether there may not be a theorem about the predicate calculus asserting that, if a formula is provable (or deducible from other formulas), it is provable (or deducible) in a certain direct fashion; in other words, a theorem giving a normal form for proofs and deductions, the proofs and deduction in normal form being in some sense direct. (End of Kleene excerpt)

There is such a theorem, which was proven by Kleene.

Formal proofs in H of even quite elementary theorems tend to be long. As a price for having analyzed logical deduction into simple steps, more of those steps have to be used. The proofs of Metamath are fully detailed formal proofs. We wish to have a means of writing rigorously verifiable mathematical proofs in a more direct fashion. Natural Deduction is a system for proving theorems and deductions in a more direct fashion. However, Natural Deduction is not compatible for use with Metamath, which uses a Hilbert-type system. Instead, Kleene's classical system G1 may be used for proving Metamath deductions and theorems in a more direct fashion.

The system of Metamath is an H system, not a Gentzen system. Therefore, proofs in Kleene's classical system G1 ("G1") cannot be included in Metamath's system H, which we shall henceforth call "system H" or "H". However, we may translate proofs in G1 into proofs in H.

By Kleene's THEOREM 47 (page 446)

if 𝜑 in G1 then 𝜑 in H

By Kleene's COROLLARY of THEOREM 47 (page 448)

if 𝜑 𝜓 in G1 then (   𝜑   ▶   𝜓   ) in H
if 𝜑   ,   𝜓 𝜒 in G1 then (   (   𝜑   ,   𝜓   )   ▶   𝜒   ) in H
if 𝜑   ,   𝜓   ,   𝜒 𝜃 in G1 then (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) in H

   ▶    denotes the same connective denoted by . " , " , in the context of Virtual Deduction, denotes the same connective denoted by . This Virtual Deduction notation is specified by the following set.mm definitions:

df-vd1 40784 ((   𝜑   ▶   𝜓   ) ↔ (𝜑𝜓))
dfvd2an 40796 ((   (   𝜑   ,   𝜓   )   ▶   𝜒   ) ((𝜑𝜓) → 𝜒))
dfvd3an 40808 ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ((𝜑𝜓𝜒) → 𝜃))

   ▶    replaces in the analogue in H of a sequent in G1 having a nonempty antecedent. If    ▶    occurs as the outermost connective denoted by    ▶    or and occurs exactly once, we call the analogue in H of a sequent in G1 a "virtual deduction" because the corresponding of the sequent is assigned properties similar to .

While sequent calculus proofs (proofs in G1) may have as steps sequents with 0, 1, or more formulas in the succedent, we shall only prove in G1 using sequents with exactly 1 formula in the succedent.

The User proves in G1 in order to obtain the benefits of more direct proving using sequent calculus, then translates the proof in G1 into a proof in H. The reference theorems and deductions to be used for proving in G1 are translations of theorems and deductions in set.mm.

Each theorem 𝜑 in set.mm corresponds to the theorem 𝜑 in G1. Deductions in G1 corresponding to deductions in H are similarly determined. Theorems in H with one or more occurrences of either    ▶    or may also be translated into theorems in G1 for by replacing the outermost occurrence of    ▶    or of the theorem in H with . Deductions in H may be translated into deductions in G1 in a similar manner. The only theorems and deductions in H useful for proving in G1 for the purpose of obtaining proofs in H are those in which, for each hypothesis or assertion, there are 0 or 1 occurrences of    ▶    and it is the outermost occurrence of    ▶    or . Kleene's THEOREM 46 and its COROLLARY 2 are used for translating from H to G1. By Kleene's THEOREM 46 (page 445)

if 𝜑 in H then 𝜑 in G1

By Kleene's COROLLARY 2 of THEOREM 46 (page 446)

if (   𝜑   ▶   𝜓   ) in H then 𝜑 𝜓 in G1
if (   (   𝜑   ,   𝜓   )   ▶   𝜒   ) in H then 𝜑   ,   𝜓 𝜒 in G1
if (   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) in H then 𝜑   ,   𝜓   ,   𝜒 𝜃 in G1

To prove in H, the User simply proves in G1 and translates each G1-proof step into a H-proof step. The translation is trivial and immediate. The proof in H is in Virtual Deduction notation. It is a working proof in the sense that, if it has no errors, each theorem and deduction of the proof is true, but may or may not, after being translated into conventional notation, unify with any theorem or deduction scheme in set.mm. Each theorem or deduction scheme in set.mm has a particular form. The working proof written by the User (the "User's Proof" or "Virtual Deduction Proof") may contain theorems and deductions which would unify with a variant of a theorem or deduction scheme in set.mm, but not with any particular form of that theorem or deduction scheme in set.mm.

The computer program completeusersproof.c may be applied to a Virtual Deduction proof to automatically add steps to the proof ("technical steps") which, if possible, transforms the form of a theorem or deduction of the Virtual Deduction proof not unifiable with a theorem or deduction scheme in set.mm into a variant form which is. For theorems and deductions of the Virtual Deduction proof which are completable in this way, completeusersproof saves the User the extra work involved in satisfying the constraint that the theorem or deduction is in a form which unifies with a theorem or deduction scheme in set.mm. mmj2, which is invoked by completeusersproof, automatically finds one of the reference theorems or deductions in set.mm which unifies with each theorem and deduction in the proof satisfying this constraint and labels the theorem or the assertion step of the deduction.

The analogs in H of the postulates of G1 are the set.mm postulates. The postulates in G1 corresponding to the Metamath postulates are not the classical system G1 postulates of Kleene (pages 442 and 443). set.mm has the predicate calculus postulates and other posulates. The Kleene classical system G1 postulates correspond to predicate calculus postulates which differ from the Metamath system G1 postulates corresponding to the predicate calculus postulates of Metamath's system H. Metamath's predicate calculus G1 postulates are presumably deducible from the Kleene classical G1 postulates and the Kleene classical G1 postulates are deducible from Metamath's G1 postulates. It should be recognized that, because of the different postulates, the classical G1 system corresponding to Metamath's system H is not identical to Kleene's classical system G1.

Why not create a separate database (setg.mm) of proofs in G1, avoiding the need to translate from H to G1 and from G1 back to H? The translations are trivial. Sequents make the language more complex than is necessary. More direct proving using sequent calculus may be done as a means towards the end of constructing proofs in H. Then, the language may be kept as simple as possible. A system G1 database would be redundant because it would duplicate the information contained in the corresponding system H database.

For earlier proofs, each "User's Proof" in the web page description of a Virtual Deduction proof in set.mm is the analogue in H of the User's working proof in G1. The User's Proof is automatically completed by completeusersproof.cmd (superseded by completeusersproof.c in September of 2016). The completed proof is the Virtual Deduction proof, which is the analogue in H of the corresponding fully detailed proof in G1. The completed Virtual Deduction proof of these earlier proofs may be automatically translated into a conventional Metamath proof.

The input for completeusersproof.c is a Virtual Deduction proof. Unlike completeusersproof.cmd, the completed proof is in conventional notation. completeusersproof.c eliminates the virtual deduction notation of the Virtual Deduction proof after utilizing the information it provides.

Applying mmj2's unify command is essential to completeusersproof. The mmj2 program is invoked within the completeusersproof.c function mmj2Unify(). The original mmj2 program was written by Mel L. O'Cat. Mario Carneiro has enhanced it. mmj2Unify() is called multiple times during the execution of completeusersproof.

A Virtual Deduction proof is a Metamath-specific version of a Natural Deduction Proof. In order for mmj2 to complete a Virtual Deduction proof it is necessary that each theorem or deduction of the proof is in a form which unifies with a theorem or deduction scheme in set.mm. completeusersproof weakens this constraint.

The User may write a Virtual Deduction proof and automatically transform it into a complete Metamath proof using the completeusersproof tool. The completed proof has been checked by the Metamath program. The task of writing a complete Metamath proof is reduced to writing what is essentially a Natural Deduction Proof.

The completeusersproof program and all associated files necessary to use it may be downloaded from the Metamath web site. All syntax definitions, theorems, and deductions necessary to create Virtual Deduction proofs are contained in set.mm. Examples of Virtual Deduction proofs in mmj2 Proof Worksheet .txt format are included in the completeusersproof download.

https://us.metamath.org/other/completeusersproof/suctrvd.html 40808, https://us.metamath.org/other/completeusersproof/sineq0altvd.html 40808, https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 40808, https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 40808, and https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 40808 are examples of Virtual Deduction proofs.

Generally, proving using Virtual Deduction and completeusersproof reduces the amount of Metamath-specific knowledge required by the User. Often, no knowledge of the specific theorems and deductions in set.mm is required to write some of the subproofs of a Virtual Deduction proof. Often, no knowledge of the Metamath-specific names of reference theorems and deductions in set.mm is required for writing some of the subproofs of a User's Proof. Often, the User may write subproofs of a proof using theorems or deductions commonly used in mathematics and correctly assume that some form of each is contained in set.mm and that completeusersproof will automatically generate the technical steps necessary to utilize them to complete the subproofs. Often, the fraction of the work which may be considered tedious is reduced and the total amount of work is reduced.

wff (   𝜑   ▶   𝜓   )
 
20.35.5  Virtual Deduction Theorems
 
Definitiondf-vd1 40784 Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.)
((   𝜑   ▶   𝜓   ) ↔ (𝜑𝜓))
 
Theoremin1 40785 Inference form of df-vd1 40784. Virtual deduction introduction rule of converting the virtual hypothesis of a 1-virtual hypothesis virtual deduction into an antecedent. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )       (𝜑𝜓)
 
Theoremiin1 40786 in1 40785 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (𝜑𝜓)
 
Theoremdfvd1ir 40787 Inference form of df-vd1 40784 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (   𝜑   ▶   𝜓   )
 
Theoremidn1 40788 Virtual deduction identity rule which is id 22 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜑   )
 
Theoremdfvd1imp 40789 Left-to-right part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((   𝜑   ▶   𝜓   ) → (𝜑𝜓))
 
Theoremdfvd1impr 40790 Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (   𝜑   ▶   𝜓   ))
 
Syntaxwvd2 40791 Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.)
wff (   𝜑   ,   𝜓   ▶   𝜒   )
 
Definitiondf-vd2 40792 Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)
((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ ((𝜑𝜓) → 𝜒))
 
Theoremdfvd2 40793 Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ (𝜑 → (𝜓𝜒)))
 
Syntaxwvhc2 40794 Syntax for a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)
wff (   𝜑   ,   𝜓   )
 
Definitiondf-vhc2 40795 Definition of a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)
((   𝜑   ,   𝜓   ) ↔ (𝜑𝜓))
 
Theoremdfvd2an 40796 Definition of a 2-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((   (   𝜑   ,   𝜓   )   ▶   𝜒   ) ↔ ((𝜑𝜓) → 𝜒))
 
Theoremdfvd2ani 40797 Inference form of dfvd2an 40796. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,   𝜓   )   ▶   𝜒   )       ((𝜑𝜓) → 𝜒)
 
Theoremdfvd2anir 40798 Right-to-left inference form of dfvd2an 40796. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (   (   𝜑   ,   𝜓   )   ▶   𝜒   )
 
Theoremdfvd2i 40799 Inference form of dfvd2 40793. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (𝜑 → (𝜓𝜒))
 
Theoremdfvd2ir 40800 Right-to-left inference form of dfvd2 40793. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (   𝜑   ,   𝜓   ▶   𝜒   )
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