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Theorem dveeq1-o 2187
 Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2018 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o x x = y → (y = zx y = z))
Distinct variable group:   x,z

Proof of Theorem dveeq1-o
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1616 . 2 (w = zx w = z)
2 ax-17 1616 . 2 (y = zw y = z)
3 equequ1 1684 . 2 (w = y → (w = zy = z))
41, 2, 3dvelimf-o 2180 1 x x = y → (y = zx y = z))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142 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:  ax11inda2ALT  2198
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