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Theorem dvelim 1890
 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 1889. Other variants of this theorem are dvelimf 1888 (with no distinct variable restrictions) and dvelimALT 1883 (that avoids ax-10 1393). (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 1416 . 2 (ψzψ)
3 dvelim.2 . 2 (z = y → (φψ))
41, 2, 3dvelimf 1888 1 x x = y → (ψxψ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1240 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by: (None)
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