Theorem zfcndext 4965

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

Assertion

Ref	Expression
zfcndext	|- (A.z(z e. x <-> z e. y) -> x = y)

Distinct variable group:   x,y,z

Proof of Theorem zfcndext

Step	Hyp	Ref	Expression
1		axextnd 4943	. . 3 |- E.z((z e. x <-> z e. y) -> x = y)
2	1	19.35i 1076	. 2 |- (A.z(z e. x <-> z e. y) -> E.z x = y)
3		19.9v 1284	. 2 |- (E.z x = y <-> x = y)
4	2, 3	sylib 198	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  E.wex 980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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