Theorem el.OLD 5406

Description: Obsolete version of el 5405 as of 6-Apr-2026. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
el.OLD	⊢ ∃𝑦 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem el.OLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pr 5390	. 2 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)
2		pm4.25 919	. . . . . 6 ⊢ (𝑧 = 𝑥 ↔ (𝑧 = 𝑥 ∨ 𝑧 = 𝑥))
3	2	imbi1i 352	. . . . 5 ⊢ ((𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦))
4	3	albii 1852	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦))
5		elequ1 2152	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
6	5	equsalvw 2037	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
7	4, 6	bitr3i 280	. . 3 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
8	7	exbii 1881	. 2 ⊢ (∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦 𝑥 ∈ 𝑦)
9	1, 8	mpbi 233	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813

This theorem is used by: (None)