Theorem zfext2 1087

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 group(s):   x,z   y,z

Proof of Theorem zfext2

Step	Hyp	Ref	Expression
1		a9e 809	. 2 ⊢ ∃w w = x
2		ax-ext 1074	. . . 4 ⊢ (∀z(z ∈ w ↔ z ∈ y) → w = y)
3		a14b 820	. . . . . . 7 ⊢ (w = x → (z ∈ w ↔ z ∈ x))
4	3	bibi1d 471	. . . . . 6 ⊢ (w = x → ((z ∈ w ↔ z ∈ y) ↔ (z ∈ x ↔ z ∈ y)))
5	4	bialdv 935	. . . . 5 ⊢ (w = x → (∀z(z ∈ w ↔ z ∈ y) ↔ ∀z(z ∈ x ↔ z ∈ y)))
6		a8b 817	. . . . 5 ⊢ (w = x → (w = y ↔ x = y))
7	5, 6	imbi12d 474	. . . 4 ⊢ (w = x → ((∀z(z ∈ w ↔ z ∈ y) → w = y) ↔ (∀z(z ∈ x ↔ z ∈ y) → x = y)))
8	2, 7	mpbii 168	. . 3 ⊢ (w = x → (∀z(z ∈ x ↔ z ∈ y) → x = y))
9	8	19.23aiv 952	. 2 ⊢ (∃w w = x → (∀z(z ∈ x ↔ z ∈ y) → x = y))
10	1, 9	ax-mp 6	1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803

This theorem is referenced by:  axextnd 3737

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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