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Theorem dveeq2 1678
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveeq2 x x = y → (z = yx z = y))
Distinct variable group:   x,z

Proof of Theorem dveeq2
StepHypRef Expression
1 ax-i12 1379 . . . . 5 (x x = z (x x = y x(z = yx z = y)))
2 orcom 634 . . . . . 6 ((x x = y x(z = yx z = y)) ↔ (x(z = yx z = y) x x = y))
32orbi2i 666 . . . . 5 ((x x = z (x x = y x(z = yx z = y))) ↔ (x x = z (x(z = yx z = y) x x = y)))
41, 3mpbi 133 . . . 4 (x x = z (x(z = yx z = y) x x = y))
5 orass 671 . . . 4 (((x x = z x(z = yx z = y)) x x = y) ↔ (x x = z (x(z = yx z = y) x x = y)))
64, 5mpbir 134 . . 3 ((x x = z x(z = yx z = y)) x x = y)
7 orel2 632 . . 3 x x = y → (((x x = z x(z = yx z = y)) x x = y) → (x x = z x(z = yx z = y))))
86, 7mpi 15 . 2 x x = y → (x x = z x(z = yx z = y)))
9 ax16 1676 . . 3 (x x = z → (z = yx z = y))
10 sp 1382 . . 3 (x(z = yx z = y) → (z = yx z = y))
119, 10jaoi 623 . 2 ((x x = z x(z = yx z = y)) → (z = yx z = y))
128, 11syl 14 1 x x = y → (z = yx z = y))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 616  wal 1226
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 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628
This theorem is referenced by:  nd5  1681  ax11v2  1683  dveeq1  1877
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