Theorem zfcndext 3759

Description: Axiom of Extensionality, reproved from conditionless ZFC version. We use only predicate calculus in the proof.

Assertion

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

Distinct variable group(s):   x,y,z

Proof of Theorem zfcndext

Step	Hyp	Ref	Expression
1		axextnd 3737	. . 3 ⊢ ∃z((z ∈ x ↔ z ∈ y) → x = y)
2	1	19.35i 755	. 2 ⊢ (∀z(z ∈ x ↔ z ∈ y) → ∃z x = y)
3		19.9rv 941	. 2 ⊢ (x = y ↔ ∃z x = y)
4	2, 3	sylibr 175	1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

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

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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