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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wl-luk-pm2.18d 34701 | Deduction based on reductio ad absurdum. Copy of pm2.18d 127 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (¬ 𝜓 → 𝜓)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | wl-luk-con4i 34702 | Inference rule. Copy of con4i 114 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜓 → 𝜑) | ||
Theorem | wl-luk-pm2.24i 34703 | Inference rule. Copy of pm2.24i 153 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜑 ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||
Theorem | wl-luk-a1i 34704 | Inference rule. Copy of a1i 11 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜑 ⇒ ⊢ (𝜓 → 𝜑) | ||
Theorem | wl-luk-mpi 34705 | A nested modus ponens inference. Copy of mpi 20 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜓 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | wl-luk-imim2i 34706 | Inference adding common antecedents in an implication. Copy of imim2i 16 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜓)) | ||
Theorem | wl-luk-imtrdi 34707 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 35 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃)) | ||
Theorem | wl-luk-ax3 34708 | ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) | ||
Theorem | wl-luk-ax1 34709 | ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | wl-luk-pm2.27 34710 | This theorem, called "Assertion", can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. Copy of pm2.27 42 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | wl-luk-com12 34711 | Inference that swaps (commutes) antecedents in an implication. Copy of com12 32 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜓 → (𝜑 → 𝜒)) | ||
Theorem | wl-luk-pm2.21 34712 | From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 123 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | ||
Theorem | wl-luk-con1i 34713 | A contraposition inference. Copy of con1i 149 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → 𝜓) ⇒ ⊢ (¬ 𝜓 → 𝜑) | ||
Theorem | wl-luk-ja 34714 | Inference joining the antecedents of two premises. Copy of ja 188 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → 𝜒) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜒) | ||
Theorem | wl-luk-imim2 34715 | A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) | ||
Theorem | wl-luk-a1d 34716 | Deduction introducing an embedded antecedent. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||
Theorem | wl-luk-ax2 34717 | ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | wl-luk-id 34718 | Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. Copy of id 22 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝜑) | ||
Theorem | wl-luk-notnotr 34719 | Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true; in intuitionistic logic, when this is true for some 𝜑, then 𝜑 is stable. Copy of notnotr 132 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Theorem | wl-luk-pm2.04 34720 | Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Copy of pm2.04 90 with a different proof. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | wl-section-impchain 34721 |
An implication like (𝜓 → 𝜑) with one antecedent can easily be
extended by prepending more and more antecedents, as in
(𝜒
→ (𝜓 → 𝜑)) or (𝜃 → (𝜒 → (𝜓 → 𝜑))). I
call these expressions implication chains, and the number of antecedents
(number of nodes minus one) denotes their length. A given length often
marks just a required minimum value, since the consequent 𝜑 itself
may represent an implication, or even an implication chain, such hiding
part of the whole chain. As an extension, it is useful to consider a
single variable 𝜑 as a degenerate implication chain of
length zero.
Implication chains play a particular role in logic, as all propositional expressions turn out to be convertible to one or more implication chains, their nodes as simple as a variable, or its negation. So there is good reason to focus on implication chains as a sort of normalized expressions, and build some general theorems around them, with proofs using recursive patterns. This allows for theorems referring to longer and longer implication chains in an automated way. The theorem names in this section contain the text fragment 'impchain' to point out their relevance to implication chains, followed by a number indicating the (minimal) length of the longest chain involved. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | wl-impchain-mp-x 34722 | This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, the theorems ax-mp 5, syl 17, syl6 35, syl8 76 form the beginning of this series. These theorems are replicated here, but with proofs that aim at a recursive scheme, allowing to base a proof on that of the previous one in the series. (Contributed by Wolf Lammen, 17-Nov-2019.) |
⊢ ⊤ | ||
Theorem | wl-impchain-mp-0 34723 |
This theorem is the start of a proof recursion scheme where we replace
the consequent of an implication chain. The number '0' in the theorem
name indicates that the modified chain has no antecedents.
This theorem is in fact a copy of ax-mp 5, and is repeated here to emphasize the recursion using similar theorem names. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜓 & ⊢ (𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | wl-impchain-mp-1 34724 | This theorem is in fact a copy of wl-luk-syl 34699, and repeated here to demonstrate a recursive proof scheme. The number '1' in the theorem name indicates that a chain of length 1 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜒 → 𝜓) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜒 → 𝜑) | ||
Theorem | wl-impchain-mp-2 34725 | This theorem is in fact a copy of wl-luk-imtrdi 34707, and repeated here to demonstrate a recursive proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜃 → (𝜒 → 𝜓)) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜃 → (𝜒 → 𝜑)) | ||
Theorem | wl-impchain-com-1.x 34726 |
It is often convenient to have the antecedent under focus in first
position, so we can apply immediate theorem forms (as opposed to
deduction, tautology form). This series of theorems swaps the first with
the last antecedent in an implication chain. This kind of swapping is
self-inverse, whence we prefer it over, say, rotating theorems. A
consequent can hide a tail of a longer chain, so theorems of this series
appear as swapping a pair of antecedents with fixed offsets. This form of
swapping antecedents is flexible enough to allow for any permutation of
antecedents in an implication chain.
The first elements of this series correspond to com12 32, com13 88, com14 96 and com15 101 in the main part. The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-mp-x 34722 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.) |
⊢ ⊤ | ||
Theorem | wl-impchain-com-1.1 34727 |
A degenerate form of antecedent swapping. The number '1' in the theorem
name indicates that it handles a chain of length 1.
Since there is just one antecedent in the chain, there is nothing to swap. Nondegenerated forms begin with wl-impchain-com-1.2 34728, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜓 → 𝜑) | ||
Theorem | wl-impchain-com-1.2 34728 |
This theorem is in fact a copy of wl-luk-com12 34711, and repeated here to
demonstrate a simple proof scheme. The number '2' in the theorem name
indicates that a chain of length 2 is modified.
See wl-impchain-com-1.x 34726 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) | ||
Theorem | wl-impchain-com-1.3 34729 |
This theorem is in fact a copy of com13 88, and repeated here to
demonstrate a simple proof scheme. The number '3' in the theorem name
indicates that a chain of length 3 is modified.
See wl-impchain-com-1.x 34726 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜑))) | ||
Theorem | wl-impchain-com-1.4 34730 |
This theorem is in fact a copy of com14 96, and repeated here to
demonstrate a simple proof scheme. The number '4' in the theorem name
indicates that a chain of length 4 is modified.
See wl-impchain-com-1.x 34726 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑)))) ⇒ ⊢ (𝜓 → (𝜃 → (𝜒 → (𝜂 → 𝜑)))) | ||
Theorem | wl-impchain-com-n.m 34731 |
This series of theorems allow swapping any two antecedents in an
implication chain. The theorem names follow a pattern wl-impchain-com-n.m
with integral numbers n < m, that swaps the m-th antecedent with n-th
one
in an implication chain. It is sufficient to restrict the length of the
chain to m, too, since the consequent can be assumed to be the tail right
of the m-th antecedent of any arbitrary sized implication chain. We
further assume n > 1, since the wl-impchain-com-1.x 34726 series already
covers the special case n = 1.
Being able to swap any two antecedents in an implication chain lays the foundation of permuting its antecedents arbitrarily. The proofs of this series aim at automated proofing using a simple scheme. Any instance of this series is a triple step of swapping the first and n-th antecedent, then the first and the m-th, then the first and the n-th antecedent again. Each of these steps is an instance of the wl-impchain-com-1.x 34726 series. (Contributed by Wolf Lammen, 17-Nov-2019.) |
⊢ ⊤ | ||
Theorem | wl-impchain-com-2.3 34732 | This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 34731. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜃 → (𝜓 → (𝜒 → 𝜑))) | ||
Theorem | wl-impchain-com-2.4 34733 | This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 34731. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑)))) ⇒ ⊢ (𝜂 → (𝜓 → (𝜒 → (𝜃 → 𝜑)))) | ||
Theorem | wl-impchain-com-3.2.1 34734 | This theorem is in fact a copy of com3r 87. The proof is an example of how to arrive at arbitrary permutations of antecedents, using only swapping theorems. The recursion principle is to first swap the correct antecedent to the position just before the consequent, and then employ a theorem handling an implication chain of length one less to reorder the others. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜓 → (𝜃 → (𝜒 → 𝜑))) | ||
Theorem | wl-impchain-a1-x 34735 |
If an implication chain is assumed (hypothesis) or proven (theorem) to
hold, then we may add any extra antecedent to it, without changing its
truth. This is expressed in its simplest form in wl-luk-a1i 34704, that
allows us prepending an arbitrary antecedent to an implication chain.
Using our antecedent swapping theorems described in
wl-impchain-com-n.m 34731, we may then move such a prepended
antecedent to
any desired location within all antecedents. The first series of theorems
of this kind adds a single antecedent somewhere to an implication chain.
The appended number in the theorem name indicates its position within all
antecedents, 1 denoting the head position. A second theorem series
extends this idea to multiple additions (TODO).
Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions than before, which renders the implication chain less versatile. So you find this proof technique mostly when you adjust a chain to a hypothesis of a rule. A common case are syllogisms merging two implication chains into one. The first elements of the first series correspond to a1i 11, a1d 25 and a1dd 50 in the main part. The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-com-1.x 34726 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.) |
⊢ ⊤ | ||
Theorem | wl-impchain-a1-1 34736 | Inference rule, a copy of a1i 11. Head start of a recursive proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜑 ⇒ ⊢ (𝜓 → 𝜑) | ||
Theorem | wl-impchain-a1-2 34737 | Inference rule, a copy of a1d 25. First recursive proof based on the previous instance. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||
Theorem | wl-impchain-a1-3 34738 | Inference rule, a copy of a1dd 50. A recursive proof depending on previous instances, and demonstrating the proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) | ||
Axiom | ax-wl-13v 34739* |
A version of ax13v 2387 with a distinctor instead of a distinct
variable
expression.
Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1907. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | wl-ax13lem1 34740* | A version of ax-wl-13v 34739 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
Theorem | wl-mps 34741 | Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜃) | ||
Theorem | wl-syls1 34742 | Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜓 → 𝜒) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜃) | ||
Theorem | wl-syls2 34743 | Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 → 𝜒) → 𝜃) | ||
Theorem | wl-embant 34744 | A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜒) | ||
Theorem | wl-orel12 34745 | In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.) |
⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒)) | ||
Theorem | wl-cases2-dnf 34746 | A particular instance of orddi 1006 and anddi 1007 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1042, and is related to consensus 1047. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1058 and dfifp4 1061, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||
Theorem | wl-cbvmotv 34747* | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.) |
⊢ (∃*𝑥⊤ → ∃*𝑦⊤) | ||
Theorem | wl-moteq 34748 | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.) |
⊢ (∃*𝑥⊤ → 𝑦 = 𝑧) | ||
Theorem | wl-motae 34749 | Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.) |
⊢ (∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧) | ||
Theorem | wl-moae 34750* | Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 1966 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 34751 and exists1 2744. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2618, trut 1539) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant ⊤. (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.) |
⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | wl-euae 34751* | Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.) |
⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | wl-nax6im 34752* | The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like 𝑥 has no instance to assign to. An expression like 𝑥 = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 1960 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 891, for example. Whatever it is, we start out with the contraposition of ax-6 1966, that guarantees the existence of at least one set. Our hypothesis here expresses tentatively it might not hold. We can simplify the antecedent then, to the point where we do not need equation any more. This suggests what a decent characterization of the empty domain of discourse could be. (Contributed by Wolf Lammen, 12-Mar-2023.) |
⊢ (¬ ∃𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∃𝑥⊤ → 𝜑) | ||
Theorem | wl-hbae1 34753 | This specialization of hbae 2449 does not depend on ax-11 2157. (Contributed by Wolf Lammen, 8-Aug-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | ||
Theorem | wl-naevhba1v 34754* | An instance of hbn1w 2049 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | wl-spae 34755 |
Prove an instance of sp 2178 from ax-13 2386 and Tarski's FOL only, without
distinct variable conditions. The antecedent ∀𝑥𝑥 = 𝑦 holds in a
multi-object universe only if 𝑦 is substituted for 𝑥, or
vice
versa, i.e. both variables are effectively the same. The converse
¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are
distinct, and it so
provides a simple translation of a distinct variable condition to a
logical term. In case studies ∀𝑥𝑥 = 𝑦 and ¬
∀𝑥𝑥 = 𝑦 can
help eliminating distinct variable conditions.
The antecedent ∀𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'. Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2173. Note that this theorem is also provable from ax-12 2173 alone, so you can pick the axiom it is based on. Compare this result to 19.3v 1982 and spaev 2053 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | wl-speqv 34756* | Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2178 is provable from Tarski's FOL and ax13v 2387 only. Note that this reverts the implication in ax13lem1 2388, so in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) |
⊢ (¬ 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | ||
Theorem | wl-19.8eqv 34757* | Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2176 is provable from Tarski's FOL and ax13v 2387 only. Note that this reverts the implication in ax13lem2 2390, so in fact (¬ 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) |
⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)) | ||
Theorem | wl-19.2reqv 34758* | Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1977 is provable from Tarski's FOL and ax13v 2387 only. Note that in conjunction with 19.2 1977 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) |
⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
Theorem | wl-nfalv 34759* | If 𝑥 is not present in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.) |
⊢ Ⅎ𝑥∀𝑦𝜑 | ||
Theorem | wl-nfimf1 34760 | An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1893 in dvelimdf 2467 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.) |
⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓)) | ||
Theorem | wl-nfae1 34761 | Unlike nfae 2451, this specialized theorem avoids ax-11 2157. (Contributed by Wolf Lammen, 26-Jun-2019.) |
⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑥 | ||
Theorem | wl-nfnae1 34762 | Unlike nfnae 2452, this specialized theorem avoids ax-11 2157. (Contributed by Wolf Lammen, 27-Jun-2019.) |
⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥 | ||
Theorem | wl-aetr 34763 | A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)) | ||
Theorem | wl-axc11r 34764 | Same as axc11r 2382, but using ax12 2441 instead of ax-12 2173 directly. This better reflects axiom usage in theorems dependent on it. (Contributed by NM, 25-Jul-2015.) Avoid direct use of ax-12 2173. (Revised by Wolf Lammen, 30-Mar-2024.) |
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | wl-dral1d 34765 | A version of dral1 2457 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, ∀𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 34777 and nf5di 2289 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) | ||
Theorem | wl-cbvalnaed 34766 | wl-cbvalnae 34767 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) & ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | wl-cbvalnae 34767 | A more general version of cbval 2412 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2466, nfsb2 2518 or dveeq1 2394. (Contributed by Wolf Lammen, 4-Jun-2019.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) & ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | wl-exeq 34768 | The semantics of ∃𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.) |
⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧)) | ||
Theorem | wl-aleq 34769 | The semantics of ∀𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.) |
⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))) | ||
Theorem | wl-nfeqfb 34770 | Extend nfeqf 2395 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.) |
⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)) | ||
Theorem | wl-nfs1t 34771 | If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2523. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | wl-equsalvw 34772* |
Version of equsalv 2264 with a disjoint variable condition, and of equsal 2435
with two disjoint variable conditions, which requires fewer axioms. See
also the dual form equsexvw 2007.
This theorem lays the foundation to a transformation of expressions called substitution of set variables in a wff. Only in this particular context we additionally assume 𝜑 and 𝑦 disjoint, stated here as 𝜑(𝑥). Similarly the disjointness of 𝜓 and 𝑥 is expressed by 𝜓(𝑦). Both 𝜑 and 𝜓 may still depend on other set variables, but that is irrelevant here. We want to transform 𝜑(𝑥) into 𝜓(𝑦) such that 𝜓 depends on 𝑦 the same way as 𝜑 depends on 𝑥. This dependency is expressed in our hypothesis (called implicit substitution): (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)). For primitive enough 𝜑 a sort of textual substitution of 𝑥 by 𝑦 is sufficient for such transformation. But note: 𝜑 must not contain wff variables, and the substitution is no proper textual substitution either. We still need grammar information to not accidently replace the x in a token 'x.' denoting multiplication, but only catch set variables 𝑥. Our current stage of development allows only equations and quantifiers make up such primitives. Thanks to equequ1 2028 and cbvalvw 2039 we can then prove in a mechanical way that in fact the implicit substitution holds for each instance. If 𝜑 contains wff variables we cannot use textual transformation any longer, since we don't know how to replace 𝑦 for 𝑥 in placeholders of unknown structure. Our theorem now states, that the generic expression ∀𝑥(𝑥 = 𝑦 → 𝜑) formally behaves as if such a substitution was possible and made. (Contributed by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | wl-equsald 34773 | Deduction version of equsal 2435. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) | ||
Theorem | wl-equsal 34774 | A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 34773 first, and then deriving more specialized versions wl-equsal 34774 and wl-equsal1t 34775 then is more efficient than the other way round, which is possible, too. See also equsal 2435. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | wl-equsal1t 34775 |
The expression 𝑥 = 𝑦 in antecedent position plays an
important role in
predicate logic, namely in implicit substitution. However, occasionally
it is irrelevant, and can safely be dropped. A sufficient condition for
this is when 𝑥 (or 𝑦 or both) is not free in
𝜑.
This theorem is more fundamental than equsal 2435, spimt 2400 or sbft 2266, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | ||
Theorem | wl-equsalcom 34776 | This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑦 = 𝑥 → 𝜑)) | ||
Theorem | wl-equsal1i 34777 | The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.) |
⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) & ⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | wl-sb6rft 34778 | A specialization of wl-equsal1t 34775. Closed form of sb6rf 2487. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))) | ||
Theorem | wl-cbvalsbi 34779* | Change bounded variables in a special case. The reverse direction seems to involve ax-11 2157. My hope is that I will in some future be able to prove mo3 2644 with reversed quantifiers not using ax-11 2157. See also the remark in mo4 2646, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.) |
⊢ (∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | wl-sbrimt 34780 | Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2309. (Contributed by Wolf Lammen, 26-Jul-2019.) |
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))) | ||
Theorem | wl-sblimt 34781 | Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2309. (Contributed by Wolf Lammen, 26-Jul-2019.) |
⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))) | ||
Theorem | wl-sb8t 34782 | Substitution of variable in universal quantifier. Closed form of sb8 2555. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-sb8et 34783 | Substitution of variable in universal quantifier. Closed form of sb8e 2556. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-sbhbt 34784 | Closed form of sbhb 2559. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) | ||
Theorem | wl-sbnf1 34785 | Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2373. Note: This theorem shows that sbnf2 2373 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) | ||
Theorem | wl-equsb3 34786 | equsb3 2105 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)) | ||
Theorem | wl-equsb4 34787 | Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧)) | ||
Theorem | wl-2sb6d 34788 | Version of 2sb6 2090 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.) |
⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑥) & ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤) & ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧) & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧) ⇒ ⊢ (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜓))) | ||
Theorem | wl-sbcom2d-lem1 34789* | Lemma used to prove wl-sbcom2d 34791. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.) |
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))) | ||
Theorem | wl-sbcom2d-lem2 34790* | Lemma used to prove wl-sbcom2d 34791. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑))) | ||
Theorem | wl-sbcom2d 34791 | Version of sbcom2 2164 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.) |
⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑤) & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧) & ⊢ (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦) ⇒ ⊢ (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓)) | ||
Theorem | wl-sbalnae 34792 | A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.) |
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | wl-sbal1 34793* | A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 34792 now. See also sbal1 2568. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | wl-sbal2 34794* | Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 34792 now. See also sbal2 2569. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | wl-2spsbbi 34795 | spsbbi 2074 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.) |
⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓)) | ||
Theorem | wl-lem-exsb 34796* | This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | wl-lem-nexmo 34797 | This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.) |
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑧)) | ||
Theorem | wl-lem-moexsb 34798* |
The antecedent ∀𝑥(𝜑 → 𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is
better suited for usage in proofs. Note that no distinct variable
restriction is placed on 𝜑.
This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.) |
⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑)) | ||
Theorem | wl-alanbii 34799 | This theorem extends alanimi 1813 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒)) | ||
Theorem | wl-mo2df 34800 | Version of mof 2643 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦) & ⊢ (𝜑 → Ⅎ𝑦𝜓) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))) |
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