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Theorem dvelim 2429
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 2427.

Other variants of this theorem are dvelimh 2428 (with no distinct variable restrictions) and dvelimhw 2321 (that avoids ax-13 2343). (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 1889 . 2 (𝜓 → ∀𝑧𝜓)
3 dvelim.2 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
41, 2, 3dvelimh 2428 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1763  df-nf 1767
This theorem is referenced by:  dvelimv  2430  axc14  2442  eujustALT  2614
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