Theorem zfcndext 10192

Description: Axiom of Extensionality ax-ext 2708, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 10170	. 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
2	1	19.36iv 1955	1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541   = wceq 1543   ∈ wcel 2112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-clel 2809  df-nfc 2879

This theorem is referenced by: (None)