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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbeqalbi | Structured version Visualization version GIF version |
Description: When both 𝑥 and 𝑧 and 𝑦 and 𝑧 are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.) |
Ref | Expression |
---|---|
sbeqalbi | ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtrr 1936 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦)) | |
2 | 1 | alrimiv 1842 | . 2 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦)) |
3 | sbeqal1 37620 | . 2 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → 𝑥 = 𝑦) | |
4 | 2, 3 | impbii 198 | 1 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: (None) |
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