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Theorem dveeq2or 1680
Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1679 but connecting xx = y by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
Assertion
Ref Expression
dveeq2or (x x = y x z = y)
Distinct variable group:   x,z

Proof of Theorem dveeq2or
StepHypRef Expression
1 ax-i12 1380 . . . . . 6 (x x = z (x x = y x(z = yx z = y)))
2 orass 671 . . . . . 6 (((x x = z x x = y) x(z = yx z = y)) ↔ (x x = z (x x = y x(z = yx z = y))))
31, 2mpbir 134 . . . . 5 ((x x = z x x = y) x(z = yx z = y))
4 pm1.4 633 . . . . . 6 ((x x = z x x = y) → (x x = y x x = z))
54orim1i 664 . . . . 5 (((x x = z x x = y) x(z = yx z = y)) → ((x x = y x x = z) x(z = yx z = y)))
63, 5ax-mp 7 . . . 4 ((x x = y x x = z) x(z = yx z = y))
7 orass 671 . . . 4 (((x x = y x x = z) x(z = yx z = y)) ↔ (x x = y (x x = z x(z = yx z = y))))
86, 7mpbi 133 . . 3 (x x = y (x x = z x(z = yx z = y)))
9 ax16 1677 . . . . . 6 (x x = z → (z = yx z = y))
109a5i 1418 . . . . 5 (x x = zx(z = yx z = y))
11 id 19 . . . . 5 (x(z = yx z = y) → x(z = yx z = y))
1210, 11jaoi 623 . . . 4 ((x x = z x(z = yx z = y)) → x(z = yx z = y))
1312orim2i 665 . . 3 ((x x = y (x x = z x(z = yx z = y))) → (x x = y x(z = yx z = y)))
148, 13ax-mp 7 . 2 (x x = y x(z = yx z = y))
15 df-nf 1330 . . . 4 (Ⅎx z = yx(z = yx z = y))
1615biimpri 124 . . 3 (x(z = yx z = y) → Ⅎx z = y)
1716orim2i 665 . 2 ((x x = y x(z = yx z = y)) → (x x = y x z = y))
1814, 17ax-mp 7 1 (x x = y x z = y)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616  wal 1226  wnf 1329
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1629
This theorem is referenced by:  equs5or  1694  sbal1yz  1860  copsexg  3954
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