Theorem elOLD

Description: Obsolete version of el as of 6-Apr-2026. (Contributed by NM, 4-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	elOLD	⊢ ∃ 𝑦 𝑥 ∈ 𝑦

Proof

Step	Hyp	Ref	Expression
1		ax-pr	⊢ ∃ 𝑦 ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 )
2		pm4.25	⊢ ( 𝑧 = 𝑥 ↔ ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) )
3	2	imbi1i	⊢ ( ( 𝑧 = 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 ) )
4	3	albii	⊢ ( ∀ 𝑧 ( 𝑧 = 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 ) )
5		elequ1	⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) )
6	5	equsalvw	⊢ ( ∀ 𝑧 ( 𝑧 = 𝑥 → 𝑧 ∈ 𝑦 ) ↔ 𝑥 ∈ 𝑦 )
7	4 6	bitr3i	⊢ ( ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ 𝑥 ∈ 𝑦 )
8	7	exbii	⊢ ( ∃ 𝑦 ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑦 𝑥 ∈ 𝑦 )
9	1 8	mpbi	⊢ ∃ 𝑦 𝑥 ∈ 𝑦