Theorem zfcndext 4945

Description: Axiom of Extensionality, reproved from conditionless ZFC version and predicate calculus.

Assertion

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

Distinct variable group:   x,y,z

Proof of Theorem zfcndext

Step	Hyp	Ref	Expression
1		axextnd 4923	. . 3 ⊢ ∃z((z ∈ x ↔ z ∈ y) → x = y)
2	1	19.35i 1074	. 2 ⊢ (∀z(z ∈ x ↔ z ∈ y) → ∃z x = y)
3		19.9v 1282	. 2 ⊢ (∃z x = y ↔ x = y)
4	2, 3	sylib 198	1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

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