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Theorem dvdemo2 5301
Description: Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑧 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑦).

That theorem bundles the theorems (𝑥(𝑥 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑥𝑧𝑥) with 𝑥, 𝑧 disjoint) and (𝑥(𝑥 = 𝑧𝑧𝑥) with 𝑥, 𝑧 disjoint).

Compare with dvdemo1 5300, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5300 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5301 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑦 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ax-10 2141, ax-11 2158, ax-12 2175, ax-13 2374. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

Assertion
Ref Expression
dvdemo2 𝑥(𝑥 = 𝑦𝑧𝑥)
Distinct variable group:   𝑥,𝑧

Proof of Theorem dvdemo2
StepHypRef Expression
1 elALT2 5296 . 2 𝑥 𝑧𝑥
2 ax-1 6 . 2 (𝑧𝑥 → (𝑥 = 𝑦𝑧𝑥))
31, 2eximii 1843 1 𝑥(𝑥 = 𝑦𝑧𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-pow 5292
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787
This theorem is referenced by: (None)
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