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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 37959. (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 3007 | . . 3 β’ (π = π β (π β π β π β π )) | |
2 | oveq1 7369 | . . . . 5 β’ (π = π β (π β¨ π ) = (π β¨ π )) | |
3 | 2 | breq2d 5122 | . . . 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 7377 | . . . . 5 β’ (π = π β ((π β¨ π ) β¨ π) = ((π β¨ π ) β¨ π)) |
6 | 5 | breq2d 5122 | . . . 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 1437 | . 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 481 | 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 397 β§ w3a 1088 = wceq 1542 β wne 2944 class class class wbr 5110 βcfv 6501 (class class class)co 7362 lecple 17147 joincjn 18207 Atomscatm 37754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-ov 7365 |
This theorem is referenced by: 3dim1 37959 |
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