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Theorem dvelim 2486
 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 2484. Other variants of this theorem are dvelimh 2485 (with no distinct variable restrictions) and dvelimhw 2337 (that avoids ax-13 2407). (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
dvelim.1 (𝜑 → ∀𝑥𝜑)
dvelim.2 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelim (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Distinct variable group:   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dvelim
StepHypRef Expression
1 dvelim.1 . 2 (𝜑 → ∀𝑥𝜑)
2 ax-5 1990 . 2 (𝜓 → ∀𝑧𝜓)
3 dvelim.2 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
41, 2, 3dvelimh 2485 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857 This theorem is referenced by:  dvelimv  2487  axc14  2518  eujustALT  2620
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