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Theorem dvelimfv 1869
Description: Like dvelimf 1873 but with a distinct variable constraint on x and z. (Contributed by Jim Kingdon, 6-Mar-2018.)
Hypotheses
Ref Expression
dvelimfv.1 (φxφ)
dvelimfv.2 (ψzψ)
dvelimfv.3 (z = y → (φψ))
Assertion
Ref Expression
dvelimfv x x = y → (ψxψ))
Distinct variable group:   x,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem dvelimfv
StepHypRef Expression
1 nfnae 1592 . . . 4 z ¬ x x = y
2 ax-i12 1379 . . . . . . . . 9 (x x = z (x x = y x(z = yx z = y)))
3 orcom 634 . . . . . . . . . 10 ((x x = y x(z = yx z = y)) ↔ (x(z = yx z = y) x x = y))
43orbi2i 666 . . . . . . . . 9 ((x x = z (x x = y x(z = yx z = y))) ↔ (x x = z (x(z = yx z = y) x x = y)))
52, 4mpbi 133 . . . . . . . 8 (x x = z (x(z = yx z = y) x x = y))
6 orass 671 . . . . . . . 8 (((x x = z x(z = yx z = y)) x x = y) ↔ (x x = z (x(z = yx z = y) x x = y)))
75, 6mpbir 134 . . . . . . 7 ((x x = z x(z = yx z = y)) x x = y)
8 nfae 1589 . . . . . . . . . . 11 xx x = z
9 ax16ALT 1721 . . . . . . . . . . 11 (x x = z → (z = yx z = y))
108, 9nfd 1397 . . . . . . . . . 10 (x x = z → Ⅎx z = y)
11 dvelimfv.1 . . . . . . . . . . . 12 (φxφ)
1211nfi 1331 . . . . . . . . . . 11 xφ
1312a1i 9 . . . . . . . . . 10 (x x = z → Ⅎxφ)
1410, 13nfimd 1459 . . . . . . . . 9 (x x = z → Ⅎx(z = yφ))
15 df-nf 1330 . . . . . . . . . 10 (Ⅎx z = yx(z = yx z = y))
16 id 19 . . . . . . . . . . 11 (Ⅎx z = y → Ⅎx z = y)
1712a1i 9 . . . . . . . . . . 11 (Ⅎx z = y → Ⅎxφ)
1816, 17nfimd 1459 . . . . . . . . . 10 (Ⅎx z = y → Ⅎx(z = yφ))
1915, 18sylbir 125 . . . . . . . . 9 (x(z = yx z = y) → Ⅎx(z = yφ))
2014, 19jaoi 623 . . . . . . . 8 ((x x = z x(z = yx z = y)) → Ⅎx(z = yφ))
2120orim1i 664 . . . . . . 7 (((x x = z x(z = yx z = y)) x x = y) → (Ⅎx(z = yφ) x x = y))
227, 21ax-mp 7 . . . . . 6 (Ⅎx(z = yφ) x x = y)
23 orcom 634 . . . . . 6 ((Ⅎx(z = yφ) x x = y) ↔ (x x = y x(z = yφ)))
2422, 23mpbi 133 . . . . 5 (x x = y x(z = yφ))
2524ori 629 . . . 4 x x = y → Ⅎx(z = yφ))
261, 25nfald 1625 . . 3 x x = y → Ⅎxz(z = yφ))
27 dvelimfv.2 . . . . 5 (ψzψ)
28 dvelimfv.3 . . . . 5 (z = y → (φψ))
2927, 28equsalh 1596 . . . 4 (z(z = yφ) ↔ ψ)
3029nfbii 1342 . . 3 (Ⅎxz(z = yφ) ↔ Ⅎxψ)
3126, 30sylib 127 . 2 x x = y → Ⅎxψ)
3231nfrd 1394 1 x x = y → (ψxψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  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-in1 532  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  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628
This theorem is referenced by: (None)
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