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Mirrors > Home > MPE Home > Th. List > spaev | Structured version Visualization version GIF version |
Description: A special instance of sp 2041
applied to an equality with a dv condition.
Unlike the more general sp 2041, we can prove this without ax-12 2034.
Instance of aeveq 1969.
The antecedent ∀𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term. Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition ∀𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.) |
Ref | Expression |
---|---|
spaev | ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ1 1939 | . 2 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦)) | |
2 | 1 | spw 1954 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: aevlem0 1967 axc11nlemOLD2 1975 |
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