P. 421 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

bi13imp23 42001

3imp 1109 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi13imp2 42002

Similar to 3imp 1109 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi12imp3 42003

Similar to 3imp 1109 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi23imp1 42004

Similar to 3imp 1109 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

bi123imp0 42005

Similar to 3imp 1109 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

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

Theorem

4animp1 42006

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.)

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

Theorem

4an31 42007

A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)

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

Theorem

4an4132 42008

A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)

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

Theorem

expcomdg 42009

Biconditional form of expcomd 416. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)

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

20.36.3  Conventional Metamath proofs, some derived from VD proofs

Theorem

iidn3 42010

idn3 42124 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 → (𝜒 → 𝜒)))

Theorem

ee222 42011

e222 42145 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 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜂))

Theorem

ee3bir 42012

Right-biconditional form of e3 42246 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 → (𝜒 → 𝜃)))    &   ⊢ (𝜏 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Theorem

ee13 42013

e13 42257 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.)

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

Theorem

ee121 42014

e121 42165 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

ee122 42015

e122 42162 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

ee333 42016

e333 42242 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

ee323 42017

e323 42275 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

3ornot23 42018

If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 42356. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Theorem

orbi1r 42019

orbi1 914 with order of disjuncts reversed. Derived from orbi1rVD 42357. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

Theorem

3orbi123 42020

pm4.39 973 with a 3-conjunct antecedent. This proof is 3orbi123VD 42359 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))

Theorem

syl5imp 42021

Closed form of syl5 34. Derived automatically from syl5impVD 42372. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

impexpd 42022

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::	⊢ (((𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜓 → (𝜒 → 𝜃)))
qed:1:	⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))

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

Theorem

com3rgbi 42023

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 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))
2::	⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃))))
3:1,2:	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃))))
4::	⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))
5::	⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃))))
6:4,5:	⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃))))
qed:3,6:	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) ↔ (𝜒 → (𝜑 → (𝜓 → 𝜃))))

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

Theorem

impexpdcom 42024

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 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
2::	⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
qed:1,2:	⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))

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

Theorem

ee1111 42025

Non-virtual deduction form of e1111 42184. (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)
h2::	⊢ (𝜑 → 𝜒)
h3::	⊢ (𝜑 → 𝜃)
h4::	⊢ (𝜑 → 𝜏)
h5::	⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
6:1,5:	⊢ (𝜑 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
7:6:	⊢ (𝜒 → (𝜑 → (𝜃 → (𝜏 → 𝜂))))
8:2,7:	⊢ (𝜑 → (𝜑 → (𝜃 → (𝜏 → 𝜂))))
9:8:	⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂)))
10:9:	⊢ (𝜃 → (𝜑 → (𝜏 → 𝜂)))
11:3,10:	⊢ (𝜑 → (𝜑 → (𝜏 → 𝜂)))
12:11:	⊢ (𝜑 → (𝜏 → 𝜂))
13:12:	⊢ (𝜏 → (𝜑 → 𝜂))
14:4,13:	⊢ (𝜑 → (𝜑 → 𝜂))
qed:14:	⊢ (𝜑 → 𝜂)

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

Theorem

pm2.43bgbi 42026

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 → (𝜑 → 𝜒))) → (𝜑 → (𝜑 → (𝜓 → 𝜒))))
2::	⊢ ((𝜑 → (𝜑 → (𝜓 → 𝜒))) → (𝜑 → (𝜓 → 𝜒)))
3:1,2:	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜑 → (𝜓 → 𝜒)))
4::	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
5:3,4:	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒)))
6::	⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → (𝜑 → 𝜒))))
qed:5,6:	⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) ↔ (𝜓 → (𝜑 → 𝜒)))

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

Theorem

pm2.43cbi 42027

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 → (𝜑 → 𝜃))) ) → (𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃)))))
2::	⊢ ((𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃))))
3:1,2:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃))))
4::	⊢ ((𝜓 → (𝜑 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃))))
5:3,4:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜒 → (𝜑 → 𝜃))))
6::	⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃)))))
qed:5,6:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))

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

Theorem

ee233 42028

Non-virtual deduction form of e233 42274. (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 → 𝜒))
h2::	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
h3::	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
h4::	⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))
5:1,4:	⊢ (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜁))) )
6:5:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) )
7:2,6:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
8:7:	⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
9:8:	⊢ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) )
10:9:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜂 → 𝜁))) )
11:10:	⊢ (𝜂 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) )
12:3,11:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
13:12:	⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
14:13:	⊢ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) )
qed:14:	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

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

Theorem

imbi13 42029

Join three logical equivalences to form equivalence of implications. imbi13 42029 is imbi13VD 42383 without virtual deductions and was automatically derived from imbi13VD 42383 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.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Theorem

ee33 42030

Non-virtual deduction form of e33 42243. (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 → (𝜒 → 𝜃)))
h2::	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
h3::	⊢ (𝜃 → (𝜏 → 𝜂))
4:1,3:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
5:4:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
6:2,5:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
7:6:	⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
8:7:	⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
qed:8:	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

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

Theorem

con5 42031

Biconditional contraposition variation. This proof is con5VD 42409 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

Theorem

con5i 42032

Inference form of con5 42031. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

exlimexi 42033

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 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)

Theorem

sb5ALT 42034*

Equivalence for substitution. Alternate proof of sb5 2271. This proof is sb5ALTVD 42422 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Theorem

eexinst01 42035

exinst01 42134 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ∃𝑥𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → 𝜒)

Theorem

eexinst11 42036

exinst11 42135 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → 𝜒)

Theorem

vk15.4j 42037

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 42037 is vk15.4jVD 42423 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(𝜃 ∧ 𝜏))    &   ⊢ ¬ ∀𝑥(𝜏 → 𝜑)    ⇒   ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Theorem

notnotrALT 42038

Converse of double negation. Alternate proof of notnotr 130. This proof is notnotrALTVD 42424 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (¬ ¬ 𝜑 → 𝜑)

Theorem

con3ALT2 42039

Contraposition. Alternate proof of con3 153. This proof is con3ALTVD 42425 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theorem

ssralv2 42040*

Quantification restricted to a subclass for two quantifiers. ssralv 3983 for two quantifiers. The proof of ssralv2 42040 was automatically generated by minimizing the automatically translated proof of ssralv2VD 42375. 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.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑))

Theorem

sbc3or 42041

sbcor 3764 with a 3-disjuncts. This proof is sbc3orgVD 42360 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](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))

Theorem

alrim3con13v 42042*

Closed form of alrimi 2209 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 42361 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))

Theorem

rspsbc2 42043*

rspsbc 3808 with two quantifying variables. This proof is rspsbc2VD 42364 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Theorem

sbcoreleleq 42044*

Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 42368. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))

Theorem

tratrb 42045*

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 42370. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)

Theorem

ordelordALT 42046

An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6273 using the Axiom of Regularity indirectly through dford2 9308. 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 42046 is ordelordALTVD 42376 without virtual deductions and was automatically derived from ordelordALTVD 42376 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 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Theorem

sbcim2g 42047

Distribution of class substitution over a left-nested implication. Similar to sbcimg 3762. sbcim2g 42047 is sbcim2gVD 42384 without virtual deductions and was automatically derived from sbcim2gVD 42384 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.)

⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Theorem

sbcbi 42048

Implication form of sbcbii 3772. sbcbi 42048 is sbcbiVD 42385 without virtual deductions and was automatically derived from sbcbiVD 42385 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.)

⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Theorem

trsbc 42049*

Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 42049 is trsbcVD 42386 without virtual deductions and was automatically derived from trsbcVD 42386 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.)

⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))

Theorem

truniALT 42050*

The union of a class of transitive sets is transitive. Alternate proof of truni 5201. truniALT 42050 is truniALTVD 42387 without virtual deductions and was automatically derived from truniALTVD 42387 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.)

⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Theorem

onfrALTlem5 42051*

Lemma for onfrALT 42058. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏 (𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))

Theorem

onfrALTlem4 42052*

Lemma for onfrALT 42058. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

Theorem

onfrALTlem3 42053*

Lemma for onfrALT 42058. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))

Theorem

ggen31 42054*

gen31 42130 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))

Theorem

onfrALTlem2 42055*

Lemma for onfrALT 42058. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))

Theorem

cbvexsv 42056*

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)

Theorem

onfrALTlem1 42057*

Lemma for onfrALT 42058. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))

Theorem

onfrALT 42058

The membership relation is foundational on the class of ordinal numbers. onfrALT 42058 is an alternate proof of onfr 6290. onfrALTVD 42400 is the Virtual Deduction proof from which onfrALT 42058 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6290 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 42400. 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

Theorem

19.41rg 42059

Closed form of right-to-left implication of 19.41 2231, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 42411. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))

Theorem

opelopab4 42060*

Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5433. (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} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Theorem

2pm13.193 42061

pm13.193 41918 for two variables. pm13.193 41918 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 42412. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Theorem

hbntal 42062

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) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Theorem

hbimpg 42063

A closed form of hbim 2299. Derived from hbimpgVD 42413. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))

Theorem

hbalg 42064

Closed form of hbal 2169. Derived from hbalgVD 42414. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

hbexg 42065

Closed form of nfex 2322. Derived from hbexgVD 42415. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Theorem

ax6e2eq 42066*

Alternate form of ax6e 2383 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 42066 is derived from ax6e2eqVD 42416. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))

Theorem

ax6e2nd 42067*

If at least two sets exist (dtru 5288) , then the same is true expressed in an alternate form similar to the form of ax6e 2383. ax6e2nd 42067 is derived from ax6e2ndVD 42417. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Theorem

ax6e2ndeq 42068*

"At least two sets exist" expressed in the form of dtru 5288 is logically equivalent to the same expressed in a form similar to ax6e 2383 if dtru 5288 is false implies 𝑢 = 𝑣. ax6e2ndeq 42068 is derived from ax6e2ndeqVD 42418. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Theorem

2sb5nd 42069*

Equivalence for double substitution 2sb5 2275 without distinct 𝑥, 𝑦 requirement. 2sb5nd 42069 is derived from 2sb5ndVD 42419. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))

Theorem

2uasbanh 42070*

Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 42070 is derived from 2uasbanhVD 42420. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))    ⇒   ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))

Theorem

2uasban 42071*

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.)

⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))

Theorem

e2ebind 42072

Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 42072 is derived from e2ebindVD 42421. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Theorem

elpwgded 42073

elpwgdedVD 42426 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜓 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵)

Theorem

trelded 42074

Deduction form of trel 5194. 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)    &   ⊢ (𝜓 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜒 → 𝐶 ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴)

Theorem

jaoded 42075

Deduction form of jao 957. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

sbtT 42076

A substitution into a theorem remains true. sbt 2070 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

Theorem

not12an2impnot1 42077

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 42077 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 42077 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.)

⊢ ((¬ (𝜑 ∧ 𝜓) ∧ 𝜓) → ¬ 𝜑)

20.36.4  What is Virtual Deduction?

Syntax

wvd1 42078

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 it 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 originally 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 his (Gentzen's) Hauptsatz or normal form theorem 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, ..., 𝜓⊢ 𝜒), 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 42079	⊢ ((   𝜑   ▶   𝜓   ) ↔ (𝜑 → 𝜓))
dfvd2an 42091	⊢ ((   (   𝜑   ,   𝜓   )   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
dfvd3an 42103	⊢ ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 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 42103, https://us.metamath.org/other/completeusersproof/sineq0altvd.html 42103, https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 42103, https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 42103, and https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 42103 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   )

20.36.5  Virtual Deduction Theorems

Definition

df-vd1 42079

Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.)

⊢ ((   𝜑   ▶   𝜓   ) ↔ (𝜑 → 𝜓))

Theorem

in1 42080

Inference form of df-vd1 42079. 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   )    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

iin1 42081

in1 42080 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

dfvd1ir 42082

Inference form of df-vd1 42079 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ (   𝜑   ▶   𝜓   )

Theorem

idn1 42083

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   )

Theorem

dfvd1imp 42084

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   ) → (𝜑 → 𝜓))

Theorem

dfvd1impr 42085

Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 → 𝜓) → (   𝜑   ▶   𝜓   ))

Syntax

wvd2 42086

Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.)

wff (   𝜑   ,   𝜓   ▶   𝜒   )

Definition

df-vd2 42087

Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)

⊢ ((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))

Theorem

dfvd2 42088

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 → (𝜓 → 𝜒)))

Syntax

wvhc2 42089

Syntax for a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)

wff (   𝜑   ,   𝜓   )

Definition

df-vhc2 42090

Definition of a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)

⊢ ((   𝜑   ,   𝜓   ) ↔ (𝜑 ∧ 𝜓))

Theorem

dfvd2an 42091

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   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))

Theorem

dfvd2ani 42092

Inference form of dfvd2an 42091. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

dfvd2anir 42093

Right-to-left inference form of dfvd2an 42091. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem

dfvd2i 42094

Inference form of dfvd2 42088. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))

Theorem

dfvd2ir 42095

Right-to-left inference form of dfvd2 42088. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )

Syntax

wvd3 42096

Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.)

wff (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )

Syntax

wvhc3 42097

Syntax for a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)

wff (   𝜑   ,   𝜓   ,   𝜒   )

Definition

df-vhc3 42098

Definition of a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)

⊢ ((   𝜑   ,   𝜓   ,   𝜒   ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))

Definition

df-vd3 42099

Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)

⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))

Theorem

dfvd3 42100

Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))