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| 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 2377. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| zfcndext | ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | axextnd 10631 | . 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | |
| 2 | 1 | 19.36iv 1946 | 1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-clel 2816 df-nfc 2892 | 
| This theorem is referenced by: (None) | 
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