MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvelim Structured version   Visualization version   GIF version

Theorem dvelim 2450
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 𝑥𝑧, conjoin them, and apply dvelimdf 2448.

Other variants of this theorem are dvelimh 2449 (with no distinct variable restrictions) and dvelimhw 2342 (that avoids ax-13 2371). Usage of this theorem is discouraged because it depends on ax-13 2371. Check out dvelimhw 2342 for a version requiring fewer axioms. (Contributed by NM, 23-Nov-1994.) (New usage is discouraged.)

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 1914 . 2 (𝜓 → ∀𝑧𝜓)
3 dvelim.2 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
41, 2, 3dvelimh 2449 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787
This theorem is referenced by:  dvelimv  2451  axc14  2462  eujustALT  2567
  Copyright terms: Public domain W3C validator