Theorem zfcndext 10651

Description: Axiom of Extensionality ax-ext 2706, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 10629	. 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
2	1	19.36iv 1944	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890

This theorem is referenced by: (None)