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Theorem 3dimlem1 33545
Description: Lemma for 3dim1 33554. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j = (join‘𝐾)
3dim0.l = (le‘𝐾)
3dim0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3dimlem1 (((𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆)) ∧ 𝑃 = 𝑄) → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)))

Proof of Theorem 3dimlem1
StepHypRef Expression
1 neeq1 2843 . . 3 (𝑃 = 𝑄 → (𝑃𝑅𝑄𝑅))
2 oveq1 6533 . . . . 5 (𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))
32breq2d 4589 . . . 4 (𝑃 = 𝑄 → (𝑆 (𝑃 𝑅) ↔ 𝑆 (𝑄 𝑅)))
43notbid 306 . . 3 (𝑃 = 𝑄 → (¬ 𝑆 (𝑃 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
52oveq1d 6541 . . . . 5 (𝑃 = 𝑄 → ((𝑃 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
65breq2d 4589 . . . 4 (𝑃 = 𝑄 → (𝑇 ((𝑃 𝑅) 𝑆) ↔ 𝑇 ((𝑄 𝑅) 𝑆)))
76notbid 306 . . 3 (𝑃 = 𝑄 → (¬ 𝑇 ((𝑃 𝑅) 𝑆) ↔ ¬ 𝑇 ((𝑄 𝑅) 𝑆)))
81, 4, 73anbi123d 1390 . 2 (𝑃 = 𝑄 → ((𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)) ↔ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))))
98biimparc 502 1 (((𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆)) ∧ 𝑃 = 𝑄) → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wne 2779   class class class wbr 4577  cfv 5789  (class class class)co 6526  lecple 15723  joincjn 16715  Atomscatm 33351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5753  df-fv 5797  df-ov 6529
This theorem is referenced by:  3dim1  33554
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