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| Mirrors > Home > MPE Home > Th. List > axextb | Structured version Visualization version GIF version | ||
| Description: A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext 2707 and df-cleq 2728. (Contributed by NM, 14-Nov-2008.) | 
| Ref | Expression | 
|---|---|
| axextb | ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elequ2g 2123 | . 2 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
| 2 | axextg 2709 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | |
| 3 | 1, 2 | impbii 209 | 1 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: dfcleq 2729 axc11next 44430 | 
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