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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3dimlem1 | Structured version Visualization version GIF version |
Description: Lemma for 3dim1 38851. (Contributed by NM, 25-Jul-2012.) |
Ref | Expression |
---|---|
3dim0.j | β’ β¨ = (joinβπΎ) |
3dim0.l | β’ β€ = (leβπΎ) |
3dim0.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
3dimlem1 | β’ (((π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π)) β§ π = π) β (π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 2997 | . . 3 β’ (π = π β (π β π β π β π )) | |
2 | oveq1 7412 | . . . . 5 β’ (π = π β (π β¨ π ) = (π β¨ π )) | |
3 | 2 | breq2d 5153 | . . . 4 β’ (π = π β (π β€ (π β¨ π ) β π β€ (π β¨ π ))) |
4 | 3 | notbid 318 | . . 3 β’ (π = π β (Β¬ π β€ (π β¨ π ) β Β¬ π β€ (π β¨ π ))) |
5 | 2 | oveq1d 7420 | . . . . 5 β’ (π = π β ((π β¨ π ) β¨ π) = ((π β¨ π ) β¨ π)) |
6 | 5 | breq2d 5153 | . . . 4 β’ (π = π β (π β€ ((π β¨ π ) β¨ π) β π β€ ((π β¨ π ) β¨ π))) |
7 | 6 | notbid 318 | . . 3 β’ (π = π β (Β¬ π β€ ((π β¨ π ) β¨ π) β Β¬ π β€ ((π β¨ π ) β¨ π))) |
8 | 1, 4, 7 | 3anbi123d 1432 | . 2 β’ (π = π β ((π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π)) β (π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π)))) |
9 | 8 | biimparc 479 | 1 β’ (((π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π)) β§ π = π) β (π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 lecple 17213 joincjn 18276 Atomscatm 38646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 |
This theorem is referenced by: 3dim1 38851 |
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