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Mirrors > Home > ILE Home > Th. List > ax9vsep | GIF version |
Description: Derive a weakened version of ax-9 1542, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4143 and Extensionality ax-ext 2171. In intuitionistic logic a9evsep 4147 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax9vsep | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9evsep 4147 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exalim 1513 | . 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1362 = wceq 1364 ∃wex 1503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-ial 1545 ax-ext 2171 ax-sep 4143 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 |
This theorem is referenced by: (None) |
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