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Theorem dvelim 2016
 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 don't require that x and y be distinct: if they aren't, 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 2082. Other variants of this theorem are dvelimh 1964 (with no distinct variable restrictions), dvelimhw 1849 (that avoids ax-12 1925), and dvelimALT 2133 (that avoids ax-10 2140). (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
dvelim.1 (φxφ)
dvelim.2 (z = y → (φψ))
Assertion
Ref Expression
dvelim x x = y → (ψxψ))
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 (φxφ)
2 ax-17 1616 . 2 (ψzψ)
3 dvelim.2 . 2 (z = y → (φψ))
41, 2, 3dvelimh 1964 1 x x = y → (ψxψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax15  2021  eujustALT  2207
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