Theorem List for Metamath Proof Explorer - 45001-45100 *Has distinct variable
group(s)
| Type | Label | Description |
|---|
| Statement |
|---|
| |
| Theorem | e233 45001 |
A virtual deduction elimination rule. (Contributed by Alan Sare,
29-Feb-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜏 ) & ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜂 ) & ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁))) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜁 ) |
| |
| Theorem | e323 45002 |
A virtual deduction elimination rule. (Contributed by Alan Sare,
17-Apr-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) & ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) & ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁))) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜁 ) |
| |
| Theorem | e000 45003 |
A virtual deduction elimination rule. The non-virtual deduction form of
e000 45003 is the virtual deduction form. (Contributed
by Alan Sare,
14-Jun-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ 𝜃 |
| |
| Theorem | e00 45004 |
Elimination rule identical to mp2 9. The non-virtual deduction form is
the virtual deduction form, which is mp2 9.
(Contributed by Alan Sare,
14-Jun-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ 𝜒 |
| |
| Theorem | e00an 45005 |
Elimination rule identical to mp2an 692. The non-virtual deduction form
is the virtual deduction form, which is mp2an 692. (Contributed by Alan
Sare, 15-Jun-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 |
| |
| Theorem | eel00cT 45006 |
An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (⊤ → 𝜒) |
| |
| Theorem | eelTT 45007 |
An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (⊤
→ 𝜑) & ⊢ (⊤
→ 𝜓) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 |
| |
| Theorem | e0a 45008 |
Elimination rule identical to ax-mp 5. The non-virtual deduction form
is the virtual deduction form, which is ax-mp 5.
(Contributed by Alan
Sare, 14-Jun-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 |
| |
| Theorem | eelT 45009 |
An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (⊤
→ 𝜑) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 |
| |
| Theorem | eel0cT 45010 |
An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (⊤ → 𝜓) |
| |
| Theorem | eelT0 45011 |
An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (⊤
→ 𝜑) & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 |
| |
| Theorem | e0bi 45012 |
Elimination rule identical to mpbi 230. The non-virtual deduction form
is the virtual deduction form, which is mpbi 230.
(Contributed by Alan
Sare, 15-Jun-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ 𝜓 |
| |
| Theorem | e0bir 45013 |
Elimination rule identical to mpbir 231. The non-virtual deduction form
is the virtual deduction form, which is mpbir 231. (Contributed by Alan
Sare, 15-Jun-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝜑 & ⊢ (𝜓 ↔ 𝜑) ⇒ ⊢ 𝜓 |
| |
| Theorem | uun0.1 45014 |
Convention notation form of un0.1 45015. (Contributed by Alan Sare,
23-Apr-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (⊤
→ 𝜑) & ⊢ (𝜓 → 𝜒)
& ⊢ ((⊤ ∧ 𝜓) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) |
| |
| Theorem | un0.1 45015 |
⊤ is the constant true, a tautology (see df-tru 1544). Kleene's
"empty conjunction" is logically equivalent to ⊤. In a virtual
deduction we shall interpret ⊤ to be the
empty wff or the empty
collection of virtual hypotheses. ⊤ in a
virtual deduction
translated into conventional notation we shall interpret to be Kleene's
empty conjunction. If 𝜃 is true given the empty collection
of
virtual hypotheses and another collection of virtual hypotheses, then it
is true given only the other collection of virtual hypotheses.
(Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( ⊤ ▶ 𝜑 ) & ⊢ ( 𝜓 ▶ 𝜒 ) & ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 )
⇒ ⊢ ( 𝜓 ▶ 𝜃 ) |
| |
| Theorem | uunT1 45016 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to
accommodate a possible future version of df-tru 1544. (Revised by David
A. Wheeler, 8-May-2019.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT1p1 45017 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT21 45018 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ (𝜑 ∧ 𝜓)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun121 45019 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun121p1 45020 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun132 45021 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun132p1 45022 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | anabss7p1 45023 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
This would have been named uun221 if the zeroth permutation did not
exist in set.mm as anabss7 673. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | un10 45024 |
A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( ( 𝜑 , ⊤ ) ▶ 𝜓 )
⇒ ⊢ ( 𝜑 ▶ 𝜓 ) |
| |
| Theorem | un01 45025 |
A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( ( ⊤ , 𝜑 ) ▶ 𝜓 )
⇒ ⊢ ( 𝜑 ▶ 𝜓 ) |
| |
| Theorem | un2122 45026 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun2131 45027 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun2131p1 45028 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uunTT1 45029 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ ⊤ ∧ 𝜑)
→ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunTT1p1 45030 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑 ∧ ⊤)
→ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunTT1p2 45031 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤ ∧ ⊤)
→ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT11 45032 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑 ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT11p1 45033 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT11p2 45034 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT12 45035 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p1 45036 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜓 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p2 45037 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p3 45038 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p4 45039 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ ⊤) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p5 45040 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun111 45041 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | 3anidm12p1 45042 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
3anidm12 1421 denotes the deduction which would have been
named uun112 if
it did not pre-exist in set.mm. This second permutation's name is based
on this pre-existing name. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | 3anidm12p2 45043 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun123 45044 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p1 45045 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p2 45046 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p3 45047 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p4 45048 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun2221 45049 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 30-Dec-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | uun2221p1 45050 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | uun2221p2 45051 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | 3impdirp1 45052 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
Commuted version of 3impdir 1352. (Contributed by Alan Sare,
4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃) |
| |
| Theorem | 3impcombi 45053 |
A 1-hypothesis propositional calculus deduction. (Contributed by Alan
Sare, 25-Sep-2017.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) |
| |
| 21.41.6 Theorems proved using Virtual
Deduction
|
| |
| Theorem | trsspwALT 45054 |
Virtual deduction proof of the left-to-right implication of dftr4 5211. A
transitive class is a subset of its power class. This proof corresponds
to the virtual deduction proof of dftr4 5211 without accumulating results.
(Contributed by Alan Sare, 29-Apr-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| |
| Theorem | trsspwALT2 45055 |
Virtual deduction proof of trsspwALT 45054. This proof is the same as the
proof of trsspwALT 45054 except each virtual deduction symbol is
replaced by
its non-virtual deduction symbol equivalent. A transitive class is a
subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| |
| Theorem | trsspwALT3 45056 |
Short predicate calculus proof of the left-to-right implication of
dftr4 5211. A transitive class is a subset of its power
class. This
proof was constructed by applying Metamath's minimize command to the
proof of trsspwALT2 45055, which is the virtual deduction proof trsspwALT 45054
without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| |
| Theorem | sspwtr 45057 |
Virtual deduction proof of the right-to-left implication of dftr4 5211. A
class which is a subclass of its power class is transitive. This proof
corresponds to the virtual deduction proof of sspwtr 45057 without
accumulating results. (Contributed by Alan Sare, 2-May-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | sspwtrALT 45058 |
Virtual deduction proof of sspwtr 45057. This proof is the same as the
proof of sspwtr 45057 except each virtual deduction symbol is
replaced by
its non-virtual deduction symbol equivalent. A class which is a
subclass of its power class is transitive. (Contributed by Alan Sare,
3-May-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | sspwtrALT2 45059 |
Short predicate calculus proof of the right-to-left implication of
dftr4 5211. A class which is a subclass of its power
class is transitive.
This proof was constructed by applying Metamath's minimize command to
the proof of sspwtrALT 45058, which is the virtual deduction proof sspwtr 45057
without virtual deductions. (Contributed by Alan Sare, 3-May-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | pwtrVD 45060 |
Virtual deduction proof of pwtr 5400; see pwtrrVD 45061 for the converse.
(Contributed by Alan Sare, 25-Aug-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr 𝒫 𝐴) |
| |
| Theorem | pwtrrVD 45061 |
Virtual deduction proof of pwtr 5400; see pwtrVD 45060 for the converse.
(Contributed by Alan Sare, 25-Aug-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ (Tr 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | suctrALT 45062 |
The successor of a transitive class is transitive. The proof of
https://us.metamath.org/other/completeusersproof/suctrvd.html
is a
Virtual Deduction proof verified by automatically transforming it into
the Metamath proof of suctrALT 45062 using completeusersproof, which is
verified by the Metamath program. The proof of
https://us.metamath.org/other/completeusersproof/suctrro.html 45062 is a
form of the completed proof which preserves the Virtual Deduction
proof's step numbers and their ordering. See suctr 6405 for the original
proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare,
12-Jun-2018.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | snssiALTVD 45063 |
Virtual deduction proof of snssiALT 45064. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
| |
| Theorem | snssiALT 45064 |
If a class is an element of another class, then its singleton is a
subclass of that other class. Alternate proof of snssi 4764. This
theorem was automatically generated from snssiALTVD 45063 using a
translation program. (Contributed by Alan Sare, 11-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
| |
| Theorem | snsslVD 45065 |
Virtual deduction proof of snssl 45066. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snssl 45066 |
If a singleton is a subclass of another class, then the singleton's
element is an element of that other class. This theorem is the
right-to-left implication of the biconditional snss 4741.
The proof of
this theorem was automatically generated from snsslVD 45065 using a tools
command file, translateMWO.cmd, by translating the proof into its
non-virtual deduction form and minimizing it. (Contributed by Alan
Sare, 25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snelpwrVD 45067 |
Virtual deduction proof of snelpwi 5392. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
| |
| Theorem | unipwrVD 45068 |
Virtual deduction proof of unipwr 45069. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | unipwr 45069 |
A class is a subclass of the union of its power class. This theorem is
the right-to-left subclass lemma of unipw 5398. The proof of this theorem
was automatically generated from unipwrVD 45068 using a tools command file ,
translateMWO.cmd , by translating the proof into its non-virtual
deduction form and minimizing it. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | sstrALT2VD 45070 |
Virtual deduction proof of sstrALT2 45071. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | sstrALT2 45071 |
Virtual deduction proof of sstr 3942, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 45070 using the command file
translate_without_overwriting.cmd . It was not minimized because the
automated minimization excluding duplicates generates a minimized proof
which, although not directly containing any duplicates, indirectly
contains a duplicate. That is, the trace back of the minimized proof
contains a duplicate. This is undesirable because some step(s) of the
minimized proof use the proven theorem. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | suctrALT2VD 45072 |
Virtual deduction proof of suctrALT2 45073. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALT2 45073 |
Virtual deduction proof of suctr 6405. The successor of a transitive
class is transitive. This proof was generated automatically from the
virtual deduction proof suctrALT2VD 45072 using the tools command file
translate_without_overwriting_minimize_excluding_duplicates.cmd .
(Contributed by Alan Sare, 11-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | elex2VD 45074* |
Virtual deduction proof of elex2 2813. (Contributed by Alan Sare,
25-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| |
| Theorem | elex22VD 45075* |
Virtual deduction proof of elex22 3465. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) |
| |
| Theorem | eqsbc2VD 45076* |
Virtual deduction proof of eqsbc2 3804. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)) |
| |
| Theorem | zfregs2VD 45077* |
Virtual deduction proof of zfregs2 9642. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ≠ ∅ → ¬
∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) |
| |
| Theorem | tpid3gVD 45078 |
Virtual deduction proof of tpid3g 4729. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
| |
| Theorem | en3lplem1VD 45079* |
Virtual deduction proof of en3lplem1 9521. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) |
| |
| Theorem | en3lplem2VD 45080* |
Virtual deduction proof of en3lplem2 9522. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) |
| |
| Theorem | en3lpVD 45081 |
Virtual deduction proof of en3lp 9523. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |
| |
| 21.41.7 Theorems proved using Virtual Deduction
with mmj2 assistance
|
| |
| Theorem | simplbi2VD 45082 |
Virtual deduction proof of simplbi2 500. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
| | 3:1,?: e0a 45008 | ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
| | qed:3,?: e0a 45008 | ⊢ (𝜓 → (𝜒 → 𝜑))
|
The proof of simplbi2 500 was automatically derived from it.
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) |
| |
| Theorem | 3ornot23VD 45083 |
Virtual deduction proof of 3ornot23 44746. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ (¬ 𝜑
∧ ¬ 𝜓) )
| | 2:: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ 𝜑 ∨ 𝜓) )
| | 3:1,?: e1a 44864 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜑 )
| | 4:1,?: e1a 44864 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜓 )
| | 5:3,4,?: e11 44925 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ (𝜑
∨ 𝜓) )
| | 6:2,?: e2 44868 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ (𝜑 ∨ 𝜓)) )
| | 7:5,6,?: e12 44960 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ 𝜒 )
| | 8:7: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ((𝜒
∨ 𝜑 ∨ 𝜓) → 𝜒) )
| | qed:8: | ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒
∨ 𝜑 ∨ 𝜓) → 𝜒))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)) |
| |
| Theorem | orbi1rVD 45084 |
Virtual deduction proof of orbi1r 44747. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜑) )
| | 3:2,?: e2 44868 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜑 ∨ 𝜒) )
| | 4:1,3,?: e12 44960 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜓 ∨ 𝜒) )
| | 5:4,?: e2 44868 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜓) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜑)
→ (𝜒 ∨ 𝜓)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜓) )
| | 8:7,?: e2 44868 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜓 ∨ 𝜒) )
| | 9:1,8,?: e12 44960 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜑 ∨ 𝜒) )
| | 10:9,?: e2 44868 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜑) )
| | 11:10: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜓)
→ (𝜒 ∨ 𝜑)) )
| | 12:6,11,?: e11 44925 | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒
∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) )
| | qed:12: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑)
↔ (𝜒 ∨ 𝜓)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))) |
| |
| Theorem | bitr3VD 45085 |
Virtual deduction proof of bitr3 352. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑
↔ 𝜓) )
| | 2:1,?: e1a 44864 | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜓
↔ 𝜑) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜑 ↔ 𝜒) )
| | 4:3,?: e2 44868 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜒 ↔ 𝜑) )
| | 5:2,4,?: e12 44960 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜓 ↔ 𝜒) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜑
↔ 𝜒) → (𝜓 ↔ 𝜒)) )
| | qed:6: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒)
→ (𝜓 ↔ 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))) |
| |
| Theorem | 3orbi123VD 45086 |
Virtual deduction proof of 3orbi123 44748. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧
(𝜏 ↔ 𝜂)) )
| | 2:1,?: e1a 44864 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜑 ↔ 𝜓) )
| | 3:1,?: e1a 44864 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜒 ↔ 𝜃) )
| | 4:1,?: e1a 44864 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜏 ↔ 𝜂) )
| | 5:2,3,?: e11 44925 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) )
| | 6:5,4,?: e11 44925 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 7:?: | ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑
∨ 𝜒 ∨ 𝜏))
| | 8:6,7,?: e10 44931 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 9:?: | ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔
(𝜓 ∨ 𝜃 ∨ 𝜂))
| | 10:8,9,?: e10 44931 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒
↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨
𝜃 ∨ 𝜂)) )
| | qed:10: | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃
∨ 𝜂)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))) |
| |
| Theorem | sbc3orgVD 45087 |
Virtual deduction proof of the analogue of sbcor 3791 with three disjuncts.
The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 3:: | ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑
∨ 𝜓 ∨ 𝜒))
| | 32:3: | ⊢ ∀𝑥(((𝜑 ∨ 𝜓) ∨ 𝜒)
↔ (𝜑 ∨ 𝜓 ∨ 𝜒))
| | 33:1,32,?: e10 44931 | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](((𝜑
∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 4:1,33,?: e11 44925 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 5:2,4,?: e11 44925 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒)) )
| | 6:1,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) )
| | 7:6,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥](𝜑
∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 8:5,7,?: e11 44925 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 9:?: | ⊢ ((([𝐴 / 𝑥]𝜑
∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑
∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))
| | 10:8,9,?: e10 44931 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓
∨ [𝐴 / 𝑥]𝜒)) )
| | qed:10: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓
∨ [𝐴 / 𝑥]𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))) |
| |
| Theorem | 19.21a3con13vVD 45088* |
Virtual deduction proof of alrim3con13v 44770. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 → ∀𝑥𝜑)
▶ (𝜑 → ∀𝑥𝜑) )
| | 2:: | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓 ∧ 𝜑
∧ 𝜒) ▶ (𝜓 ∧ 𝜑 ∧ 𝜒) )
| | 3:2,?: e2 44868 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜓 )
| | 4:2,?: e2 44868 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜑 )
| | 5:2,?: e2 44868 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜒 )
| | 6:1,4,?: e12 44960 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜑 )
| | 7:3,?: e2 44868 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜓 )
| | 8:5,?: e2 44868 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜒 )
| | 9:7,6,8,?: e222 44873 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒) )
| | 10:9,?: e2 44868 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒) )
| | 11:10:in2 | ⊢ ( (𝜑 → ∀𝑥𝜑) ▶ ((𝜓
∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)) )
| | qed:11:in1 | ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓
∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))) |
| |
| Theorem | exbirVD 45089 |
Virtual deduction proof of exbir 44716. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
▶ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) )
| | 2:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ,
(𝜑 ∧ 𝜓) ▶ (𝜑 ∧ 𝜓) )
| | 3:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ,
(𝜑 ∧ 𝜓), 𝜃 ▶ 𝜃 )
| | 5:1,2,?: e12 44960 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜒 ↔ 𝜃) )
| | 6:3,5,?: e32 44994 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃 ▶ 𝜒 )
| | 7:6: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜃 → 𝜒) )
| | 8:7: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
▶ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒)) )
| | 9:8,?: e1a 44864 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)) ▶ (𝜑 → (𝜓 → (𝜃 → 𝜒))) )
| | qed:9: | ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
→ (𝜑 → (𝜓 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 13-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) |
| |
| Theorem | exbiriVD 45090 |
Virtual deduction proof of exbiri 810. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
| | 2:: | ⊢ ( 𝜑 ▶ 𝜑 )
| | 3:: | ⊢ ( 𝜑 , 𝜓 ▶ 𝜓 )
| | 4:: | ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜃 )
| | 5:2,1,?: e10 44931 | ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) )
| | 6:3,5,?: e21 44966 | ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) )
| | 7:4,6,?: e32 44994 | ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜒 )
| | 8:7: | ⊢ ( 𝜑 , 𝜓 ▶ (𝜃 → 𝜒) )
| | 9:8: | ⊢ ( 𝜑 ▶ (𝜓 → (𝜃 → 𝜒)) )
| | qed:9: | ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
| |
| Theorem | rspsbc2VD 45091* |
Virtual deduction proof of rspsbc2 44771. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ 𝐶 ∈ 𝐷 )
| | 3:: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 )
| | 4:1,3,?: e13 44984 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑 )
| | 5:1,4,?: e13 44984 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑 )
| | 6:2,5,?: e23 44991 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑 )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ (∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) )
| | 8:7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝐶 ∈ 𝐷
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) )
| | qed:8: | ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) |
| |
| Theorem | 3impexpVD 45092 |
Virtual deduction proof of 3impexp 1359. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 2:: | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒)
↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
| | 3:1,2,?: e10 44931 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 4:3,?: e1a 44864 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 5:4,?: e1a 44864 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (𝜑 → (𝜓 → (𝜒 → 𝜃))) )
| | 6:5: | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
→ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
| | 7:: | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (𝜑 → (𝜓 → (𝜒 → 𝜃))) )
| | 8:7,?: e1a 44864 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 9:8,?: e1a 44864 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 10:2,9,?: e01 44928 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 11:10: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
| | qed:6,11,?: e00 45004 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) |
| |
| Theorem | 3impexpbicomVD 45093 |
Virtual deduction proof of 3impexpbicom 44717. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) )
| | 2:: | ⊢ ((𝜃 ↔ 𝜏) ↔ (𝜏
↔ 𝜃))
| | 3:1,2,?: e10 44931 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) )
| | 4:3,?: e1a 44864 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))) )
| | 5:4: | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))))
| | 6:: | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))) )
| | 7:6,?: e1a 44864 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏
↔ 𝜃)) )
| | 8:7,2,?: e10 44931 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏)) )
| | 9:8: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏)))
| | qed:5,9,?: e00 45004 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))) |
| |
| Theorem | 3impexpbicomiVD 45094 |
Virtual deduction proof of 3impexpbicomi 44718. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏))
| | qed:1,?: e0a 45008 | ⊢ (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃))))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) |
| |
| Theorem | sbcoreleleqVD 45095* |
Virtual deduction proof of sbcoreleleq 44772. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 ∈
𝑦 ↔ 𝑥 ∈ 𝐴) )
| | 3:1,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑦 ∈
𝑥 ↔ 𝐴 ∈ 𝑥) )
| | 4:1,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 =
𝑦 ↔ 𝑥 = 𝐴) )
| | 5:2,3,4,?: e111 44911 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ((𝑥 ∈ 𝐴
∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥
∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 6:1,?: e1a 44864 | ⊢ ( 𝐴 ∈ 𝐵
▶ ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥
∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 7:5,6: e11 44925 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦](𝑥
∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)) )
| | qed:7: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))) |
| |
| Theorem | hbra2VD 45096* |
Virtual deduction proof of nfra2 3346. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ (∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 2:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 3:1,2,?: e00 45004 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 4:2: | ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 5:4,?: e0a 45008 | ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | qed:3,5,?: e00 45004 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑)
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
| |
| Theorem | tratrbVD 45097* |
Virtual deduction proof of tratrb 44773. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)
∧ 𝐵 ∈ 𝐴) )
| | 2:1,?: e1a 44864 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 3:1,?: e1a 44864 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 4:1,?: e1a 44864 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| | 5:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) )
| | 6:5,?: e2 44868 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| | 7:5,?: e2 44868 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐵 )
| | 8:2,7,4,?: e121 44893 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐴 )
| | 9:2,6,8,?: e122 44890 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐴 )
| | 10:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ 𝐵 ∈ 𝑥 )
| | 11:6,7,10,?: e223 44872 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥) )
| | 12:11: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)) )
| | 13:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵
∧ 𝐵 ∈ 𝑥)
| | 14:12,13,?: e20 44963 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝐵 ∈ 𝑥 )
| | 15:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑥 = 𝐵 )
| | 16:7,15,?: e23 44991 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑦 ∈ 𝑥 )
| | 17:6,16,?: e23 44991 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) )
| | 18:17: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) )
| | 19:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
| | 20:18,19,?: e20 44963 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| | 21:3,?: e1a 44864 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑦 ∈ 𝐴
∀𝑥 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 22:21,9,4,?: e121 44893 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) )
| | 23:22,?: e2 44868 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 24:4,23,?: e12 44960 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵) )
| | 25:14,20,24,?: e222 44873 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐵 )
| | 26:25: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ((𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 27:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨
𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 28:27,?: e0a 45008 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
→ ∀𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| | 29:28,26: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 30:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑥∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 31:30,?: e0a 45008 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥(Tr 𝐴
∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| | 32:31,29: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥
∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 33:32,?: e1a 44864 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| | qed:33: | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
|
(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 | al2imVD 45098 |
Virtual deduction proof of al2im 1815. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ ∀𝑥(𝜑 → (𝜓 → 𝜒)) )
| | 2:1,?: e1a 44864 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| | 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| | 4:2,3,?: e10 44931 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) )
| | qed:4: | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒))
→ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))) |
| |
| Theorem | syl5impVD 45099 |
Virtual deduction proof of syl5imp 44749. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜑
→ (𝜓 → 𝜒)) )
| | 2:1,?: e1a 44864 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| | 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| | 4:3,2,?: e21 44966 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| | 5:4,?: e2 44868 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜑 → (𝜃 → 𝜒)) )
| | 6:5: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))) )
| | qed:6: | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))) |
| |
| Theorem | idiVD 45100 |
Virtual deduction proof of idiALT 44715. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ 𝜑
| | qed:1,?: e0a 45008 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 ⇒ ⊢ 𝜑 |
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