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Theorem ax16i 1714
Description: Inference with ax-16 1671 as its conclusion, that doesn't require ax-10 1372, ax-11 1373, or ax-12 1378 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 (x = z → (φψ))
ax16i.2 (ψxψ)
Assertion
Ref Expression
ax16i (x x = y → (φxφ))
Distinct variable groups:   x,y,z   φ,z
Allowed substitution hints:   φ(x,y)   ψ(x,y,z)

Proof of Theorem ax16i
StepHypRef Expression
1 ax-17 1395 . . . 4 (x = yz x = y)
2 ax-17 1395 . . . 4 (z = yx z = y)
3 ax-8 1371 . . . 4 (x = z → (x = yz = y))
41, 2, 3cbv3h 1607 . . 3 (x x = yz z = y)
5 ax-8 1371 . . . . . 6 (z = x → (z = yx = y))
65spimv 1668 . . . . 5 (z z = yx = y)
7 equid 1565 . . . . . . . 8 x = x
8 ax-8 1371 . . . . . . . 8 (x = y → (x = xy = x))
97, 8mpi 15 . . . . . . 7 (x = yy = x)
10 equid 1565 . . . . . . . . 9 z = z
11 ax-8 1371 . . . . . . . . 9 (z = y → (z = zy = z))
1210, 11mpi 15 . . . . . . . 8 (z = yy = z)
13 ax-8 1371 . . . . . . . 8 (y = z → (y = xz = x))
1412, 13syl 14 . . . . . . 7 (z = y → (y = xz = x))
159, 14syl5com 26 . . . . . 6 (x = y → (z = yz = x))
161, 15alimdh 1332 . . . . 5 (x = y → (z z = yz z = x))
176, 16mpcom 32 . . . 4 (z z = yz z = x)
18 ax-8 1371 . . . . . 6 (z = x → (z = zx = z))
1910, 18mpi 15 . . . . 5 (z = xx = z)
2019alimi 1320 . . . 4 (z z = xz x = z)
2117, 20syl 14 . . 3 (z z = yz x = z)
224, 21syl 14 . 2 (x x = yz x = z)
23 ax-17 1395 . . . 4 (φzφ)
24 ax16i.1 . . . . 5 (x = z → (φψ))
2524biimpcd 148 . . . 4 (φ → (x = zψ))
2623, 25alimdh 1332 . . 3 (φ → (z x = zzψ))
27 ax16i.2 . . . 4 (ψxψ)
2824biimprd 147 . . . . 5 (x = z → (ψφ))
2919, 28syl 14 . . . 4 (z = x → (ψφ))
3027, 23, 29cbv3h 1607 . . 3 (zψxφ)
3126, 30syl6com 31 . 2 (z x = z → (φxφ))
3222, 31syl 14 1 (x x = y → (φxφ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403
This theorem depends on definitions:  df-bi 110  df-nf 1326
This theorem is referenced by:  ax16ALT  1715
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