Theorem axext3 2805

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.) Remove dependencies on ax-10 2192, ax-12 2220, ax-13 2389. (Revised by Wolf Lammen, 9-Dec-2019.)

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 2178	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
2	1	bibi1d 335	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
3	2	albidv 2019	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
4		ax-ext 2803	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦)
5	3, 4	syl6bir 246	. . 3 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦))
6		ax7 2120	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 → 𝑥 = 𝑦))
7	5, 6	syld 47	. 2 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
8		ax6ev 2077	. 2 ⊢ ∃𝑤 𝑤 = 𝑥
9	7, 8	exlimiiv 2030	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879

This theorem is referenced by:  axext4  2807  dfcleq  2819  axextnd  9735  axextdist  32238  bj-cleqhyp  33408