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Mirrors > Home > ILE Home > Th. List > ax16 | Unicode version |
Description: Theorem showing that ax-16 1825 is redundant if ax-17 1537 is included in the
axiom system. The important part of the proof is provided by aev 1823.
See ax16ALT 1870 for an alternate proof that does not require ax-10 1516 or ax12 1523. This theorem should not be referenced in any proof. Instead, use ax-16 1825 below so that theorems needing ax-16 1825 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Ref | Expression |
---|---|
ax16 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 1823 |
. 2
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2 | ax-17 1537 |
. . . 4
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3 | sbequ12 1782 |
. . . . 5
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4 | 3 | biimpcd 159 |
. . . 4
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5 | 2, 4 | alimdh 1478 |
. . 3
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6 | 2 | hbsb3 1819 |
. . . 4
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7 | stdpc7 1781 |
. . . 4
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8 | 6, 2, 7 | cbv3h 1754 |
. . 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-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: dveeq2 1826 dveeq2or 1827 a16g 1875 exists2 2135 |
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