Description: Principle of transitive induction. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015) (Revised by AV, 13-Jul-2024)
Ref | Expression | ||
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Hypotheses | rtrclind.1 | |
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rtrclind.2 | |
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rtrclind.3 | |
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rtrclind.4 | |
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rtrclind.5 | |
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rtrclind.6 | |
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rtrclind.7 | |
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rtrclind.8 | |
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rtrclind.9 | |
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Assertion | rtrclind | |
Step | Hyp | Ref | Expression |
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1 | rtrclind.1 | |
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2 | rtrclind.2 | |
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3 | rtrclind.3 | |
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4 | rtrclind.4 | |
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5 | rtrclind.5 | |
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6 | rtrclind.6 | |
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7 | rtrclind.7 | |
|
8 | rtrclind.8 | |
|
9 | rtrclind.9 | |
|
10 | 1 | dfrtrcl2 | |
11 | 1 | dfrtrclrec2 | |
12 | 11 | biimpac | |
13 | simprl | |
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14 | simprrr | |
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15 | simprrl | |
|
16 | 1 2 3 4 5 6 7 8 9 | relexpind | |
17 | 13 14 15 16 | syl3c | |
18 | 17 | anassrs | |
19 | 18 | expcom | |
20 | 19 | expcom | |
21 | 20 | rexlimiv | |
22 | 12 21 | mpcom | |
23 | 22 | expcom | |
24 | breq | |
|
25 | 24 | imbi1d | |
26 | 23 25 | syl5ibr | |
27 | 10 26 | mpcom | |