Theorem axextg 2795

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 2145, ax-12 2177, ax-13 2390. (Revised by BJ, 12-Jul-2019.) (Revised by Wolf Lammen, 9-Dec-2019.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axextg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2129	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
2	1	bibi1d 346	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
3	2	albidv 1921	. . 3 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
4		equequ1 2032	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
5	3, 4	imbi12d 347	. 2 ⊢ (𝑤 = 𝑥 → ((∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)))
6		ax-ext 2793	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦)
7	5, 6	chvarvv 2005	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  axextb  2796  axextnd  10013  axextdist  33044