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Mirrors > Home > ILE Home > Th. List > dveeq1 | GIF version |
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.) |
Ref | Expression |
---|---|
dveeq1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dveeq2 1743 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
2 | equcom 1639 | . 2 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) | |
3 | 2 | albii 1404 | . 2 ⊢ (∀𝑥 𝑧 = 𝑦 ↔ ∀𝑥 𝑦 = 𝑧) |
4 | 1, 2, 3 | 3imtr3g 202 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1287 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 |
This theorem is referenced by: sbal2 1946 |
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