Theorem zfext2 1454

Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct.

Assertion

Ref	Expression
zfext2	⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

Distinct variable groups:   x,z   y,z

Proof of Theorem zfext2

Step	Hyp	Ref	Expression
1		a9e 1121	. 2 ⊢ ∃w w = x
2		ax-ext 1452	. . . 4 ⊢ (∀z(z ∈ w ↔ z ∈ y) → w = y)
3		elequ2 1133	. . . . . . 7 ⊢ (w = x → (z ∈ w ↔ z ∈ x))
4	3	bibi1d 617	. . . . . 6 ⊢ (w = x → ((z ∈ w ↔ z ∈ y) ↔ (z ∈ x ↔ z ∈ y)))
5	4	albidv 1273	. . . . 5 ⊢ (w = x → (∀z(z ∈ w ↔ z ∈ y) ↔ ∀z(z ∈ x ↔ z ∈ y)))
6		equequ1 1130	. . . . 5 ⊢ (w = x → (w = y ↔ x = y))
7	5, 6	imbi12d 624	. . . 4 ⊢ (w = x → ((∀z(z ∈ w ↔ z ∈ y) → w = y) ↔ (∀z(z ∈ x ↔ z ∈ y) → x = y)))
8	2, 7	mpbii 193	. . 3 ⊢ (w = x → (∀z(z ∈ x ↔ z ∈ y) → x = y))
9	8	19.23aiv 1290	. 2 ⊢ (∃w w = x → (∀z(z ∈ x ↔ z ∈ y) → x = y))
10	1, 9	ax-mp 7	1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 951   = wceq 953   ∈ wcel 955  ∃wex 977

This theorem is referenced by:  axextnd 4915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-8 961  ax-9 962  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978

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