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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 xz, 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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