Theorem List for Intuitionistic Logic Explorer - 2601-2700 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | rexlimdvv 2601* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Jul-2004.)
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Theorem | rexlimdvva 2602* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 18-Jun-2014.)
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Theorem | r19.26 2603 |
Theorem 19.26 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | r19.27v 2604* |
Restricted quantitifer version of one direction of 19.27 1561. (The other
direction holds when is inhabited, see r19.27mv 3519.) (Contributed
by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(Proof shortened by Wolf Lammen, 17-Jun-2023.)
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Theorem | r19.28v 2605* |
Restricted quantifier version of one direction of 19.28 1563. (The other
direction holds when is inhabited, see r19.28mv 3515.) (Contributed
by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
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Theorem | r19.26-2 2606 |
Theorem 19.26 of [Margaris] p. 90 with 2
restricted quantifiers.
(Contributed by NM, 10-Aug-2004.)
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Theorem | r19.26-3 2607 |
Theorem 19.26 of [Margaris] p. 90 with 3
restricted quantifiers.
(Contributed by FL, 22-Nov-2010.)
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Theorem | r19.26m 2608 |
Theorem 19.26 of [Margaris] p. 90 with mixed
quantifiers. (Contributed by
NM, 22-Feb-2004.)
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Theorem | ralbi 2609 |
Distribute a restricted universal quantifier over a biconditional.
Theorem 19.15 of [Margaris] p. 90 with
restricted quantification.
(Contributed by NM, 6-Oct-2003.)
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Theorem | rexbi 2610 |
Distribute a restricted existential quantifier over a biconditional.
Theorem 19.18 of [Margaris] p. 90 with
restricted quantification.
(Contributed by Jim Kingdon, 21-Jan-2019.)
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Theorem | ralbiim 2611 |
Split a biconditional and distribute quantifier. (Contributed by NM,
3-Jun-2012.)
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Theorem | r19.27av 2612* |
Restricted version of one direction of Theorem 19.27 of [Margaris]
p. 90. (The other direction doesn't hold when is empty.)
(Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | r19.28av 2613* |
Restricted version of one direction of Theorem 19.28 of [Margaris]
p. 90. (The other direction doesn't hold when is empty.)
(Contributed by NM, 2-Apr-2004.)
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Theorem | r19.29 2614 |
Theorem 19.29 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | r19.29r 2615 |
Variation of Theorem 19.29 of [Margaris] p. 90
with restricted
quantifiers. (Contributed by NM, 31-Aug-1999.)
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Theorem | ralnex2 2616 |
Relationship between two restricted universal and existential quantifiers.
(Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf
Lammen, 18-May-2023.)
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Theorem | r19.29af2 2617 |
A commonly used pattern based on r19.29 2614. (Contributed by Thierry
Arnoux, 17-Dec-2017.)
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Theorem | r19.29af 2618* |
A commonly used pattern based on r19.29 2614. (Contributed by Thierry
Arnoux, 29-Nov-2017.)
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Theorem | r19.29an 2619* |
A commonly used pattern based on r19.29 2614. (Contributed by Thierry
Arnoux, 29-Dec-2019.)
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Theorem | r19.29a 2620* |
A commonly used pattern based on r19.29 2614. (Contributed by Thierry
Arnoux, 22-Nov-2017.)
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Theorem | r19.29d2r 2621 |
Theorem 19.29 of [Margaris] p. 90 with two
restricted quantifiers,
deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
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Theorem | r19.29vva 2622* |
A commonly used pattern based on r19.29 2614, version with two restricted
quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
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Theorem | r19.32r 2623 |
One direction of Theorem 19.32 of [Margaris]
p. 90 with restricted
quantifiers. For decidable propositions this is an equivalence.
(Contributed by Jim Kingdon, 19-Aug-2018.)
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Theorem | r19.30dc 2624 |
Restricted quantifier version of 19.30dc 1627. (Contributed by Scott
Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
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DECID  
 
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Theorem | r19.32vr 2625* |
One direction of Theorem 19.32 of [Margaris]
p. 90 with restricted
quantifiers. For decidable propositions this is an equivalence, as seen
at r19.32vdc 2626. (Contributed by Jim Kingdon, 19-Aug-2018.)
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Theorem | r19.32vdc 2626* |
Theorem 19.32 of [Margaris] p. 90 with
restricted quantifiers, where
is
decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
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DECID           |
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Theorem | r19.35-1 2627 |
Restricted quantifier version of 19.35-1 1624. (Contributed by Jim Kingdon,
4-Jun-2018.)
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Theorem | r19.36av 2628* |
One direction of a restricted quantifier version of Theorem 19.36 of
[Margaris] p. 90. In classical logic,
the converse would hold if
has at least one element, but in intuitionistic logic, that is not a
sufficient condition. (Contributed by NM, 22-Oct-2003.)
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Theorem | r19.37 2629 |
Restricted version of one direction of Theorem 19.37 of [Margaris]
p. 90. In classical logic the converse would hold if has at least
one element, but that is not sufficient in intuitionistic logic.
(Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro,
11-Dec-2016.)
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Theorem | r19.37av 2630* |
Restricted version of one direction of Theorem 19.37 of [Margaris]
p. 90. (Contributed by NM, 2-Apr-2004.)
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Theorem | r19.40 2631 |
Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90.
(Contributed by NM, 2-Apr-2004.)
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Theorem | r19.41 2632 |
Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
(Contributed by NM, 1-Nov-2010.)
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Theorem | r19.41v 2633* |
Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
(Contributed by NM, 17-Dec-2003.)
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Theorem | r19.42v 2634* |
Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed
by NM, 27-May-1998.)
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Theorem | r19.43 2635 |
Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by
NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
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Theorem | r19.44av 2636* |
One direction of a restricted quantifier version of Theorem 19.44 of
[Margaris] p. 90. The other direction
doesn't hold when is
empty.
(Contributed by NM, 2-Apr-2004.)
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Theorem | r19.45av 2637* |
Restricted version of one direction of Theorem 19.45 of [Margaris]
p. 90. (The other direction doesn't hold when is empty.)
(Contributed by NM, 2-Apr-2004.)
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Theorem | ralcomf 2638* |
Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
14-Oct-2016.)
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Theorem | rexcomf 2639* |
Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
14-Oct-2016.)
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Theorem | ralcom 2640* |
Commutation of restricted quantifiers. (Contributed by NM,
13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
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Theorem | rexcom 2641* |
Commutation of restricted quantifiers. (Contributed by NM,
19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
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Theorem | rexcom13 2642* |
Swap 1st and 3rd restricted existential quantifiers. (Contributed by
NM, 8-Apr-2015.)
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Theorem | rexrot4 2643* |
Rotate existential restricted quantifiers twice. (Contributed by NM,
8-Apr-2015.)
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Theorem | ralcom3 2644 |
A commutative law for restricted quantifiers that swaps the domain of the
restriction. (Contributed by NM, 22-Feb-2004.)
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Theorem | reean 2645* |
Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
(Proof shortened by Andrew Salmon, 30-May-2011.)
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Theorem | reeanv 2646* |
Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
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Theorem | 3reeanv 2647* |
Rearrange three existential quantifiers. (Contributed by Jeff Madsen,
11-Jun-2010.)
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Theorem | nfreu1 2648 |
is not free in   .
(Contributed by NM,
19-Mar-1997.)
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Theorem | nfrmo1 2649 |
is not free in   .
(Contributed by NM,
16-Jun-2017.)
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Theorem | nfreudxy 2650* |
Not-free deduction for restricted uniqueness. This is a version where
and are distinct. (Contributed
by Jim Kingdon,
6-Jun-2018.)
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Theorem | nfreuxy 2651* |
Not-free for restricted uniqueness. This is a version where and
are distinct.
(Contributed by Jim Kingdon, 6-Jun-2018.)
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Theorem | rabid 2652 |
An "identity" law of concretion for restricted abstraction. Special
case
of Definition 2.1 of [Quine] p. 16.
(Contributed by NM, 9-Oct-2003.)
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Theorem | rabid2 2653* |
An "identity" law for restricted class abstraction. (Contributed by
NM,
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
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Theorem | rabbi 2654 |
Equivalent wff's correspond to equal restricted class abstractions.
Closed theorem form of rabbidva 2725. (Contributed by NM, 25-Nov-2013.)
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Theorem | rabswap 2655 |
Swap with a membership relation in a restricted class abstraction.
(Contributed by NM, 4-Jul-2005.)
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Theorem | nfrab1 2656 |
The abstraction variable in a restricted class abstraction isn't free.
(Contributed by NM, 19-Mar-1997.)
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Theorem | nfrabxy 2657* |
A variable not free in a wff remains so in a restricted class
abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
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Theorem | reubida 2658 |
Formula-building rule for restricted existential quantifier (deduction
form). (Contributed by Mario Carneiro, 19-Nov-2016.)
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Theorem | reubidva 2659* |
Formula-building rule for restricted existential quantifier (deduction
form). (Contributed by NM, 13-Nov-2004.)
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Theorem | reubidv 2660* |
Formula-building rule for restricted existential quantifier (deduction
form). (Contributed by NM, 17-Oct-1996.)
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Theorem | reubiia 2661 |
Formula-building rule for restricted existential quantifier (inference
form). (Contributed by NM, 14-Nov-2004.)
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Theorem | reubii 2662 |
Formula-building rule for restricted existential quantifier (inference
form). (Contributed by NM, 22-Oct-1999.)
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Theorem | rmobida 2663 |
Formula-building rule for restricted existential quantifier (deduction
form). (Contributed by NM, 16-Jun-2017.)
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Theorem | rmobidva 2664* |
Formula-building rule for restricted existential quantifier (deduction
form). (Contributed by NM, 16-Jun-2017.)
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Theorem | rmobidv 2665* |
Formula-building rule for restricted existential quantifier (deduction
form). (Contributed by NM, 16-Jun-2017.)
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Theorem | rmobiia 2666 |
Formula-building rule for restricted existential quantifier (inference
form). (Contributed by NM, 16-Jun-2017.)
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Theorem | rmobii 2667 |
Formula-building rule for restricted existential quantifier (inference
form). (Contributed by NM, 16-Jun-2017.)
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Theorem | raleqf 2668 |
Equality theorem for restricted universal quantifier, with
bound-variable hypotheses instead of distinct variable restrictions.
(Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon,
11-Jul-2011.)
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Theorem | rexeqf 2669 |
Equality theorem for restricted existential quantifier, with
bound-variable hypotheses instead of distinct variable restrictions.
(Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon,
11-Jul-2011.)
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Theorem | reueq1f 2670 |
Equality theorem for restricted unique existential quantifier, with
bound-variable hypotheses instead of distinct variable restrictions.
(Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon,
11-Jul-2011.)
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Theorem | rmoeq1f 2671 |
Equality theorem for restricted at-most-one quantifier, with
bound-variable hypotheses instead of distinct variable restrictions.
(Contributed by Alexander van der Vekens, 17-Jun-2017.)
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Theorem | raleq 2672* |
Equality theorem for restricted universal quantifier. (Contributed by
NM, 16-Nov-1995.)
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Theorem | rexeq 2673* |
Equality theorem for restricted existential quantifier. (Contributed by
NM, 29-Oct-1995.)
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Theorem | reueq1 2674* |
Equality theorem for restricted unique existential quantifier.
(Contributed by NM, 5-Apr-2004.)
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Theorem | rmoeq1 2675* |
Equality theorem for restricted at-most-one quantifier. (Contributed by
Alexander van der Vekens, 17-Jun-2017.)
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Theorem | raleqi 2676* |
Equality inference for restricted universal qualifier. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | rexeqi 2677* |
Equality inference for restricted existential qualifier. (Contributed
by Mario Carneiro, 23-Apr-2015.)
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Theorem | raleqdv 2678* |
Equality deduction for restricted universal quantifier. (Contributed by
NM, 13-Nov-2005.)
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Theorem | rexeqdv 2679* |
Equality deduction for restricted existential quantifier. (Contributed
by NM, 14-Jan-2007.)
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Theorem | raleqbi1dv 2680* |
Equality deduction for restricted universal quantifier. (Contributed by
NM, 16-Nov-1995.)
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Theorem | rexeqbi1dv 2681* |
Equality deduction for restricted existential quantifier. (Contributed
by NM, 18-Mar-1997.)
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Theorem | reueqd 2682* |
Equality deduction for restricted unique existential quantifier.
(Contributed by NM, 5-Apr-2004.)
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Theorem | rmoeqd 2683* |
Equality deduction for restricted at-most-one quantifier. (Contributed
by Alexander van der Vekens, 17-Jun-2017.)
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Theorem | raleqbidv 2684* |
Equality deduction for restricted universal quantifier. (Contributed by
NM, 6-Nov-2007.)
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Theorem | rexeqbidv 2685* |
Equality deduction for restricted universal quantifier. (Contributed by
NM, 6-Nov-2007.)
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Theorem | raleqbidva 2686* |
Equality deduction for restricted universal quantifier. (Contributed by
Mario Carneiro, 5-Jan-2017.)
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Theorem | rexeqbidva 2687* |
Equality deduction for restricted universal quantifier. (Contributed by
Mario Carneiro, 5-Jan-2017.)
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Theorem | mormo 2688 |
Unrestricted "at most one" implies restricted "at most
one". (Contributed
by NM, 16-Jun-2017.)
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Theorem | reu5 2689 |
Restricted uniqueness in terms of "at most one". (Contributed by NM,
23-May-1999.) (Revised by NM, 16-Jun-2017.)
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Theorem | reurex 2690 |
Restricted unique existence implies restricted existence. (Contributed by
NM, 19-Aug-1999.)
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Theorem | reurmo 2691 |
Restricted existential uniqueness implies restricted "at most one."
(Contributed by NM, 16-Jun-2017.)
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Theorem | rmo5 2692 |
Restricted "at most one" in term of uniqueness. (Contributed by NM,
16-Jun-2017.)
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Theorem | nrexrmo 2693 |
Nonexistence implies restricted "at most one". (Contributed by NM,
17-Jun-2017.)
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Theorem | cbvralfw 2694* |
Rule used to change bound variables, using implicit substitution.
Version of cbvralf 2696 with a disjoint variable condition. Although
we
don't do so yet, we expect this disjoint variable condition will allow
us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof.
(Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto,
23-May-2024.)
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Theorem | cbvrexfw 2695* |
Rule used to change bound variables, using implicit substitution.
Version of cbvrexf 2697 with a disjoint variable condition, which
does not
require ax-13 2150. (Contributed by FL, 27-Apr-2008.) (Revised
by Gino
Giotto, 10-Jan-2024.)
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Theorem | cbvralf 2696 |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro,
9-Oct-2016.)
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Theorem | cbvrexf 2697 |
Rule used to change bound variables, using implicit substitution.
(Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro,
9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
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Theorem | cbvralw 2698* |
Rule used to change bound variables, using implicit substitution.
Version of cbvral 2699 with a disjoint variable condition. Although
we
don't do so yet, we expect this disjoint variable condition will allow
us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof.
(Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto,
10-Jan-2024.)
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Theorem | cbvral 2699* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 31-Jul-2003.)
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Theorem | cbvrex 2700* |
Rule used to change bound variables, using implicit substitution.
(Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon,
8-Jun-2011.)
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