Theorem axextg 2710

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 2141, ax-12 2175, ax-13 2371. (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 2125	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
2	1	bibi1d 347	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
3	2	albidv 1928	. . 3 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
4		equequ1 2033	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
5	3, 4	imbi12d 348	. 2 ⊢ (𝑤 = 𝑥 → ((∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)))
6		ax-ext 2708	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦)
7	5, 6	chvarvv 2007	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788

This theorem is referenced by:  axextb  2711  axextnd  10205  axextdist  33494