Theorem dvdemo2 5240

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 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (⊢ ∃𝑥(𝑥 = 𝑥 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑧 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint).

Compare with dvdemo1 5239, 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 5239 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5240 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 2142, ax-11 2158, ax-12 2175, ax-13 2379. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

Assertion

Ref	Expression
dvdemo2	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑧

Proof of Theorem dvdemo2

Step	Hyp	Ref	Expression
1		el 5235	. 2 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-1 6	. 2 ⊢ (𝑧 ∈ 𝑥 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
3	1, 2	eximii 1838	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

This theorem is referenced by: (None)