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Theorem dveeq1 1892
 Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
dveeq1 x x = y → (y = zx y = z))
Distinct variable group:   x,z

Proof of Theorem dveeq1
StepHypRef Expression
1 dveeq2 1693 . 2 x x = y → (z = yx z = y))
2 equcom 1590 . 2 (z = yy = z)
32albii 1356 . 2 (x z = yx y = z)
41, 2, 33imtr3g 193 1 x x = y → (y = zx y = z))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  sbal2  1895
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