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Axiom ax-bnd 1373
Description: Axiom of bundling. The general idea of this axiom is that two variables are either distinct or non-distinct. That idea could be expressed as zz = x ¬ zz = x. However, we instead choose an axiom which has many of the same consequences, but which is different with respect to a universe which contains only one object. zz = x holds if z and x are the same variable, likewise for z and y, and xz(x = yzx = y) holds if z is distinct from the others (and the universe has at least two objects).

As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability).

This axiom implies ax-i12 1372 as can be seen at axi12 1381. Whether ax-bnd can be proved from the remaining axioms including ax-i12 1372 is not known.

The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.)

Assertion
Ref Expression
ax-bnd (z z = x (z z = y xz(x = yz x = y)))

Detailed syntax breakdown of Axiom ax-bnd
StepHypRef Expression
1 vz . . . 4 setvar z
2 vx . . . 4 setvar x
31, 2weq 1366 . . 3 wff z = x
43, 1wal 1222 . 2 wff z z = x
5 vy . . . . 5 setvar y
61, 5weq 1366 . . . 4 wff z = y
76, 1wal 1222 . . 3 wff z z = y
82, 5weq 1366 . . . . . 6 wff x = y
98, 1wal 1222 . . . . . 6 wff z x = y
108, 9wi 4 . . . . 5 wff (x = yz x = y)
1110, 1wal 1222 . . . 4 wff z(x = yz x = y)
1211, 2wal 1222 . . 3 wff xz(x = yz x = y)
137, 12wo 613 . 2 wff (z z = y xz(x = yz x = y))
144, 13wo 613 1 wff (z z = x (z z = y xz(x = yz x = y)))
Colors of variables: wff set class
This axiom is referenced by:  axi12  1381  nfsbxy  1792  nfsbxyt  1793  sbcomxyyz  1820  dvelimor  1868  oprabidlem  5449
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