Theorem zfcndext 10568

Description: Axiom of Extensionality ax-ext 2733, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2402. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

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 10546	. 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
2	1	19.36iv 1965	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557   = wceq 1559   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-clel 2836  df-nfc 2910

This theorem is referenced by: (None)