Theorem axext3 2160

Description: A generalization of the Axiom of Extensionality in which 𝑥 and 𝑦 need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
axext3	⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axext3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2153	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
2	1	bibi1d 233	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
3	2	albidv 1824	. . 3 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
4		equequ1 1712	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
5	3, 4	imbi12d 234	. 2 ⊢ (𝑤 = 𝑥 → ((∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)))
6		ax-ext 2159	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦)
7	5, 6	chvarv 1937	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-14 2151  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  axext4  2161