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Theorem ax16i 1783
Description: Inference with ax-16 1739 as its conclusion, that doesn't require ax-10 1439, ax-11 1440, or ax-12 1445 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 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
ax16i (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
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 1462 . . . 4 (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
2 ax-17 1462 . . . 4 (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
3 ax-8 1438 . . . 4 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
41, 2, 3cbv3h 1675 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
5 ax-8 1438 . . . . . 6 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
65spimv 1736 . . . . 5 (∀𝑧 𝑧 = 𝑦𝑥 = 𝑦)
7 equid 1632 . . . . . . . 8 𝑥 = 𝑥
8 ax-8 1438 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑥𝑦 = 𝑥))
97, 8mpi 15 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
10 equid 1632 . . . . . . . . 9 𝑧 = 𝑧
11 ax-8 1438 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 = 𝑧𝑦 = 𝑧))
1210, 11mpi 15 . . . . . . . 8 (𝑧 = 𝑦𝑦 = 𝑧)
13 ax-8 1438 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = 𝑥𝑧 = 𝑥))
1412, 13syl 14 . . . . . . 7 (𝑧 = 𝑦 → (𝑦 = 𝑥𝑧 = 𝑥))
159, 14syl5com 29 . . . . . 6 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
161, 15alimdh 1399 . . . . 5 (𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
176, 16mpcom 36 . . . 4 (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
18 ax-8 1438 . . . . . 6 (𝑧 = 𝑥 → (𝑧 = 𝑧𝑥 = 𝑧))
1910, 18mpi 15 . . . . 5 (𝑧 = 𝑥𝑥 = 𝑧)
2019alimi 1387 . . . 4 (∀𝑧 𝑧 = 𝑥 → ∀𝑧 𝑥 = 𝑧)
2117, 20syl 14 . . 3 (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
224, 21syl 14 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
23 ax-17 1462 . . . 4 (𝜑 → ∀𝑧𝜑)
24 ax16i.1 . . . . 5 (𝑥 = 𝑧 → (𝜑𝜓))
2524biimpcd 157 . . . 4 (𝜑 → (𝑥 = 𝑧𝜓))
2623, 25alimdh 1399 . . 3 (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧𝜓))
27 ax16i.2 . . . 4 (𝜓 → ∀𝑥𝜓)
2824biimprd 156 . . . . 5 (𝑥 = 𝑧 → (𝜓𝜑))
2919, 28syl 14 . . . 4 (𝑧 = 𝑥 → (𝜓𝜑))
3027, 23, 29cbv3h 1675 . . 3 (∀𝑧𝜓 → ∀𝑥𝜑)
3126, 30syl6com 35 . 2 (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
3222, 31syl 14 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  ax16ALT  1784
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