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Theorem dveeq2 1693
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 1395 . . . . 5 (x x = z (x x = y x(z = yx z = y)))
2 orcom 646 . . . . . 6 ((x x = y x(z = yx z = y)) ↔ (x(z = yx z = y) x x = y))
32orbi2i 678 . . . . 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 683 . . . 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 644 . . 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 1691 . . 3 (x x = z → (z = yx z = y))
10 sp 1398 . . 3 (x(z = yx z = y) → (z = yx z = y))
119, 10jaoi 635 . 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 628  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:  nd5  1696  ax11v2  1698  dveeq1  1892
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