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Theorem dvelim 2444
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 2442.

Other variants of this theorem are dvelimh 2443 (with no distinct variable restrictions) and dvelimhw 2335 (that avoids ax-13 2365). Usage of this theorem is discouraged because it depends on ax-13 2365. Check out dvelimhw 2335 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 1905 . 2 (𝜓 → ∀𝑧𝜓)
3 dvelim.2 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
41, 2, 3dvelimh 2443 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778
This theorem is referenced by:  dvelimv  2445  axc14  2456  eujustALT  2560
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