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Theorem dvelimALT 1883
Description: Version of dvelim 1890 that doesn't use ax-10 1393. Because it has different distinct variable constraints than dvelim 1890 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvelimALT.1 (φxφ)
dvelimALT.2 (z = y → (φψ))
Assertion
Ref Expression
dvelimALT x x = y → (ψxψ))
Distinct variable groups:   ψ,z   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y)

Proof of Theorem dvelimALT
StepHypRef Expression
1 nfv 1418 . . . 4 z ¬ x x = y
2 ax-i12 1395 . . . . . . . . 9 (x x = z (x x = y x(z = yx z = y)))
3 orcom 646 . . . . . . . . . 10 ((x x = y x(z = yx z = y)) ↔ (x(z = yx z = y) x x = y))
43orbi2i 678 . . . . . . . . 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 683 . . . . . . . 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 nfa1 1431 . . . . . . . . . . 11 xx x = z
9 ax16ALT 1736 . . . . . . . . . . 11 (x x = z → (z = yx z = y))
108, 9nfd 1413 . . . . . . . . . 10 (x x = z → Ⅎx z = y)
11 dvelimALT.1 . . . . . . . . . . . 12 (φxφ)
1211nfi 1348 . . . . . . . . . . 11 xφ
1312a1i 9 . . . . . . . . . 10 (x x = z → Ⅎxφ)
1410, 13nfimd 1474 . . . . . . . . 9 (x x = z → Ⅎx(z = yφ))
15 df-nf 1347 . . . . . . . . . 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 1474 . . . . . . . . . 10 (Ⅎx z = y → Ⅎx(z = yφ))
1915, 18sylbir 125 . . . . . . . . 9 (x(z = yx z = y) → Ⅎx(z = yφ))
2014, 19jaoi 635 . . . . . . . 8 ((x x = z x(z = yx z = y)) → Ⅎx(z = yφ))
2120orim1i 676 . . . . . . 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 646 . . . . . 6 ((Ⅎx(z = yφ) x x = y) ↔ (x x = y x(z = yφ)))
2422, 23mpbi 133 . . . . 5 (x x = y x(z = yφ))
2524ori 641 . . . 4 x x = y → Ⅎx(z = yφ))
261, 25nfald 1640 . . 3 x x = y → Ⅎxz(z = yφ))
27 ax-17 1416 . . . . 5 (ψzψ)
28 dvelimALT.2 . . . . 5 (z = y → (φψ))
2927, 28equsalh 1611 . . . 4 (z(z = yφ) ↔ ψ)
3029nfbii 1359 . . 3 (Ⅎxz(z = yφ) ↔ Ⅎxψ)
3126, 30sylib 127 . 2 x x = y → Ⅎxψ)
3231nfrd 1410 1 x x = y → (ψxψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  wal 1240  wnf 1346
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-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  hbsb4  1885
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