Theorem cleljustALT 2402

Description: Alternate proof of cleljust 2158. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how disjoint variable conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cleljustALT	⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljustALT

Step	Hyp	Ref	Expression
1		ax-5 1937	. . 3 ⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)
2		elequ1 2156	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
3	1, 2	equsexhv 2333	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
4	3	bicomi 227	1 ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811

This theorem is referenced by: (None)