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Theorem dvdemo2 5248
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 5247, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See 5247 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5248 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 2146, ax-11 2162, ax-12 2178, ax-13 2391. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

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

Proof of Theorem dvdemo2
StepHypRef Expression
1 el 5243 . 2 𝑥 𝑧𝑥
2 ax-1 6 . 2 (𝑧𝑥 → (𝑥 = 𝑦𝑧𝑥))
31, 2eximii 1838 1 𝑥(𝑥 = 𝑦𝑧𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-pow 5239
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by: (None)
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