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Mirrors > Home > MPE Home > Th. List > Mathboxes > elim2ifim | Structured version Visualization version Unicode version |
Description: Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
elim2if.1 | |
elim2if.2 | |
elim2if.3 | |
elim2ifim.1 | |
elim2ifim.2 | |
elim2ifim.3 |
Ref | Expression |
---|---|
elim2ifim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid 431 | . . 3 | |
2 | elim2ifim.1 | . . . . 5 | |
3 | 2 | ancli 574 | . . . 4 |
4 | pm4.42 1004 | . . . . . 6 | |
5 | elim2ifim.2 | . . . . . . . . . 10 | |
6 | 5 | ex 450 | . . . . . . . . 9 |
7 | 6 | ancld 576 | . . . . . . . 8 |
8 | 7 | imp 445 | . . . . . . 7 |
9 | elim2ifim.3 | . . . . . . . . . 10 | |
10 | 9 | ex 450 | . . . . . . . . 9 |
11 | 10 | ancld 576 | . . . . . . . 8 |
12 | 11 | imp 445 | . . . . . . 7 |
13 | 8, 12 | orim12i 538 | . . . . . 6 |
14 | 4, 13 | sylbi 207 | . . . . 5 |
15 | 14 | ancli 574 | . . . 4 |
16 | 3, 15 | orim12i 538 | . . 3 |
17 | 1, 16 | ax-mp 5 | . 2 |
18 | elim2if.1 | . . 3 | |
19 | elim2if.2 | . . 3 | |
20 | elim2if.3 | . . 3 | |
21 | 18, 19, 20 | elim2if 29363 | . 2 |
22 | 17, 21 | mpbir 221 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 cif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087 |
This theorem is referenced by: (None) |
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