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Mirrors > Home > MPE Home > Th. List > axext3 | Structured version Visualization version GIF version |
Description: A generalization of the Axiom of Extensionality in which 𝑥 and 𝑦 need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 2019, ax-12 2047, ax-13 2246. (Revised by Wolf Lammen, 9-Dec-2019.) |
Ref | Expression |
---|---|
axext3 | ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 2004 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)) | |
2 | 1 | bibi1d 333 | . . . . 5 ⊢ (𝑤 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) |
3 | 2 | albidv 1849 | . . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) |
4 | ax-ext 2602 | . . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦) | |
5 | 3, 4 | syl6bir 244 | . . 3 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦)) |
6 | ax7 1943 | . . 3 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 → 𝑥 = 𝑦)) | |
7 | 5, 6 | syld 47 | . 2 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)) |
8 | ax6ev 1890 | . 2 ⊢ ∃𝑤 𝑤 = 𝑥 | |
9 | 7, 8 | exlimiiv 1859 | 1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: axext4 2606 dfcleq 2616 axextnd 9413 axextdist 31705 bj-cleqhyp 32892 |
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