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Theorem ax16i 1738
 Description: Inference with ax-16 1695 as its conclusion, that doesn't require ax-10 1396, ax-11 1397, or ax-12 1402 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
Hypotheses
Ref Expression
ax16i.1
ax16i.2
Assertion
Ref Expression
ax16i
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   (,,)

Proof of Theorem ax16i
StepHypRef Expression
1 ax-17 1419 . . . 4
2 ax-17 1419 . . . 4
3 ax-8 1395 . . . 4
41, 2, 3cbv3h 1631 . . 3
5 ax-8 1395 . . . . . 6
65spimv 1692 . . . . 5
7 equid 1589 . . . . . . . 8
8 ax-8 1395 . . . . . . . 8
97, 8mpi 15 . . . . . . 7
10 equid 1589 . . . . . . . . 9
11 ax-8 1395 . . . . . . . . 9
1210, 11mpi 15 . . . . . . . 8
13 ax-8 1395 . . . . . . . 8
1412, 13syl 14 . . . . . . 7
159, 14syl5com 26 . . . . . 6
161, 15alimdh 1356 . . . . 5
176, 16mpcom 32 . . . 4
18 ax-8 1395 . . . . . 6
1910, 18mpi 15 . . . . 5
2019alimi 1344 . . . 4
2117, 20syl 14 . . 3
224, 21syl 14 . 2
23 ax-17 1419 . . . 4
24 ax16i.1 . . . . 5
2524biimpcd 148 . . . 4
2623, 25alimdh 1356 . . 3
27 ax16i.2 . . . 4
2824biimprd 147 . . . . 5
2919, 28syl 14 . . . 4
3027, 23, 29cbv3h 1631 . . 3
3126, 30syl6com 31 . 2
3222, 31syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98  wal 1241 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  ax16ALT  1739
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