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Theorem dvelim 2463
 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 2461. Other variants of this theorem are dvelimh 2462 (with no distinct variable restrictions) and dvelimhw 2356 (that avoids ax-13 2380). Usage of this theorem is discouraged because it depends on ax-13 2380. Check out dvelimhw 2356 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 1912 . 2 (𝜓 → ∀𝑧𝜓)
3 dvelim.2 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
41, 2, 3dvelimh 2462 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1537 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 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787 This theorem is referenced by:  dvelimv  2464  axc14  2476  eujustALT  2592
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