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Theorem cdleme31se2 40323
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.)
Hypotheses
Ref Expression
cdleme31se2.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))
cdleme31se2.y 𝑌 = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme31se2 (𝑆𝐴𝑆 / 𝑡𝐸 = 𝑌)
Distinct variable groups:   𝑡,𝐴   𝑡,   𝑡,   𝑡,𝑃   𝑡,𝑄   𝑡,𝑅   𝑡,𝑆   𝑡,𝑊
Allowed substitution hints:   𝐷(𝑡)   𝐸(𝑡)   𝑌(𝑡)

Proof of Theorem cdleme31se2
StepHypRef Expression
1 nfcv 2897 . . . . 5 𝑡(𝑃 𝑄)
2 nfcv 2897 . . . . 5 𝑡
3 nfcsb1v 3896 . . . . . 6 𝑡𝑆 / 𝑡𝐷
4 nfcv 2897 . . . . . 6 𝑡
5 nfcv 2897 . . . . . 6 𝑡((𝑅 𝑆) 𝑊)
63, 4, 5nfov 7429 . . . . 5 𝑡(𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))
71, 2, 6nfov 7429 . . . 4 𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
87a1i 11 . . 3 (𝑆𝐴𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
9 csbeq1a 3886 . . . . 5 (𝑡 = 𝑆𝐷 = 𝑆 / 𝑡𝐷)
10 oveq2 7407 . . . . . 6 (𝑡 = 𝑆 → (𝑅 𝑡) = (𝑅 𝑆))
1110oveq1d 7414 . . . . 5 (𝑡 = 𝑆 → ((𝑅 𝑡) 𝑊) = ((𝑅 𝑆) 𝑊))
129, 11oveq12d 7417 . . . 4 (𝑡 = 𝑆 → (𝐷 ((𝑅 𝑡) 𝑊)) = (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
1312oveq2d 7415 . . 3 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
148, 13csbiegf 3905 . 2 (𝑆𝐴𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
15 cdleme31se2.e . . 3 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))
1615csbeq2i 3880 . 2 𝑆 / 𝑡𝐸 = 𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))
17 cdleme31se2.y . 2 𝑌 = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
1814, 16, 173eqtr4g 2794 1 (𝑆𝐴𝑆 / 𝑡𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wnfc 2882  csb 3872  (class class class)co 7399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-iota 6480  df-fv 6535  df-ov 7402
This theorem is referenced by:  cdlemeg47rv2  40450
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