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Theorem dvelimor 1876
Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula φ (containing z) and a distinct variable constraint between x and z. The theorem makes it possible to replace the distinct variable constraint with the disjunct xx = y (ψ is just a version of φ with y substituted for z). (Contributed by Jim Kingdon, 11-May-2018.)
Hypotheses
Ref Expression
dvelimor.1 xφ
dvelimor.2 (z = y → (φψ))
Assertion
Ref Expression
dvelimor (x x = y xψ)
Distinct variable groups:   ψ,z   x,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y)

Proof of Theorem dvelimor
StepHypRef Expression
1 ax-bnd 1380 . . . . . 6 (x x = z (x x = y zx(z = yx z = y)))
2 orcom 634 . . . . . . 7 ((x x = y zx(z = yx z = y)) ↔ (zx(z = yx z = y) x x = y))
32orbi2i 666 . . . . . 6 ((x x = z (x x = y zx(z = yx z = y))) ↔ (x x = z (zx(z = yx z = y) x x = y)))
41, 3mpbi 133 . . . . 5 (x x = z (zx(z = yx z = y) x x = y))
5 orass 671 . . . . 5 (((x x = z zx(z = yx z = y)) x x = y) ↔ (x x = z (zx(z = yx z = y) x x = y)))
64, 5mpbir 134 . . . 4 ((x x = z zx(z = yx z = y)) x x = y)
7 nfae 1589 . . . . . . 7 zx x = z
8 a16nf 1728 . . . . . . 7 (x x = z → Ⅎx(z = yφ))
97, 8alrimi 1396 . . . . . 6 (x x = zzx(z = yφ))
10 df-nf 1330 . . . . . . . 8 (Ⅎx z = yx(z = yx z = y))
11 id 19 . . . . . . . . 9 (Ⅎx z = y → Ⅎx z = y)
12 dvelimor.1 . . . . . . . . . 10 xφ
1312a1i 9 . . . . . . . . 9 (Ⅎx z = y → Ⅎxφ)
1411, 13nfimd 1459 . . . . . . . 8 (Ⅎx z = y → Ⅎx(z = yφ))
1510, 14sylbir 125 . . . . . . 7 (x(z = yx z = y) → Ⅎx(z = yφ))
1615alimi 1324 . . . . . 6 (zx(z = yx z = y) → zx(z = yφ))
179, 16jaoi 623 . . . . 5 ((x x = z zx(z = yx z = y)) → zx(z = yφ))
1817orim1i 664 . . . 4 (((x x = z zx(z = yx z = y)) x x = y) → (zx(z = yφ) x x = y))
196, 18ax-mp 7 . . 3 (zx(z = yφ) x x = y)
20 orcom 634 . . 3 ((zx(z = yφ) x x = y) ↔ (x x = y zx(z = yφ)))
2119, 20mpbi 133 . 2 (x x = y zx(z = yφ))
22 nfalt 1452 . . . 4 (zx(z = yφ) → Ⅎxz(z = yφ))
23 ax-17 1400 . . . . . 6 (ψzψ)
24 dvelimor.2 . . . . . 6 (z = y → (φψ))
2523, 24equsalh 1596 . . . . 5 (z(z = yφ) ↔ ψ)
2625nfbii 1342 . . . 4 (Ⅎxz(z = yφ) ↔ Ⅎxψ)
2722, 26sylib 127 . . 3 (zx(z = yφ) → Ⅎxψ)
2827orim2i 665 . 2 ((x x = y zx(z = yφ)) → (x x = y xψ))
2921, 28ax-mp 7 1 (x x = y xψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628
This theorem is referenced by:  nfsb4or  1881  rgen2a  2353
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