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Mirrors > Home > MPE Home > Th. List > dvelim | Structured version Visualization version GIF version |
Description: This theorem can be used
to eliminate a distinct variable restriction on
𝑥 and 𝑧 and replace it with the
"distinctor" ¬ ∀𝑥𝑥 = 𝑦
as an antecedent. 𝜑 normally has 𝑧 free and can be read
𝜑(𝑧), and 𝜓 substitutes 𝑦 for
𝑧
and can be read
𝜑(𝑦). We do not require that 𝑥 and
𝑦
be distinct: if
they are not, the distinctor will become false (in multiple-element
domains of discourse) and "protect" the consequent.
To obtain a closed-theorem form of this inference, prefix the hypotheses with ∀𝑥∀𝑧, conjoin them, and apply dvelimdf 2335. Other variants of this theorem are dvelimh 2336 (with no distinct variable restrictions) and dvelimhw 2173 (that avoids ax-13 2246). (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
dvelim.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
dvelim.2 | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dvelim | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvelim.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | ax-5 1839 | . 2 ⊢ (𝜓 → ∀𝑧𝜓) | |
3 | dvelim.2 | . 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | dvelimh 2336 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → 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-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 |
This theorem is referenced by: dvelimv 2338 axc14 2372 eujustALT 2473 |
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