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Mirrors > Home > ILE Home > Th. List > ax16 | Unicode version |
Description: Theorem showing that ax-16 1743 is redundant if ax-17 1465 is included in the
axiom system. The important part of the proof is provided by aev 1741.
See ax16ALT 1788 for an alternate proof that does not require ax-10 1442 or ax-12 1448. This theorem should not be referenced in any proof. Instead, use ax-16 1743 below so that theorems needing ax-16 1743 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Ref | Expression |
---|---|
ax16 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 1741 |
. 2
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2 | ax-17 1465 |
. . . 4
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3 | sbequ12 1702 |
. . . . 5
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4 | 3 | biimpcd 158 |
. . . 4
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5 | 2, 4 | alimdh 1402 |
. . 3
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6 | 2 | hbsb3 1737 |
. . . 4
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7 | stdpc7 1701 |
. . . 4
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8 | 6, 2, 7 | cbv3h 1679 |
. . 3
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9 | 5, 8 | syl6com 35 |
. 2
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10 | 1, 9 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 |
This theorem is referenced by: dveeq2 1744 dveeq2or 1745 a16g 1793 exists2 2046 |
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