Theorem dvdemo1 5335

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

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 = 𝑦 → 𝑥 ∈ 𝑥) with 𝑥, 𝑦 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥) with 𝑥, 𝑦 disjoint).

Compare with dvdemo2 5336, 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 5336 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.)

Note that dvdemo1 5335 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑧 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require ax-11 2194 nor ax-13 2406. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

Assertion

Ref	Expression
dvdemo1	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem dvdemo1

Step	Hyp	Ref	Expression
1		dtruALT2 5332	. . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		exnal 1850	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbir 234	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
4		pm2.21 124	. 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
5	3, 4	eximii 1860	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-nul 5261  ax-pow 5327

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by: (None)