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Theorem cdleme31se2 34487
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 2745 . . . . 5 𝑡(𝑃 𝑄)
2 nfcv 2745 . . . . 5 𝑡
3 nfcsb1v 3509 . . . . . 6 𝑡𝑆 / 𝑡𝐷
4 nfcv 2745 . . . . . 6 𝑡
5 nfcv 2745 . . . . . 6 𝑡((𝑅 𝑆) 𝑊)
63, 4, 5nfov 6548 . . . . 5 𝑡(𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))
71, 2, 6nfov 6548 . . . 4 𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
87a1i 11 . . 3 (𝑆𝐴𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
9 csbeq1a 3502 . . . . 5 (𝑡 = 𝑆𝐷 = 𝑆 / 𝑡𝐷)
10 oveq2 6530 . . . . . 6 (𝑡 = 𝑆 → (𝑅 𝑡) = (𝑅 𝑆))
1110oveq1d 6537 . . . . 5 (𝑡 = 𝑆 → ((𝑅 𝑡) 𝑊) = ((𝑅 𝑆) 𝑊))
129, 11oveq12d 6540 . . . 4 (𝑡 = 𝑆 → (𝐷 ((𝑅 𝑡) 𝑊)) = (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
1312oveq2d 6538 . . 3 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
148, 13csbiegf 3517 . 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 3939 . 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 2663 1 (𝑆𝐴𝑆 / 𝑡𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  wnfc 2732  csb 3493  (class class class)co 6522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
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 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-iota 5749  df-fv 5793  df-ov 6525
This theorem is referenced by:  cdlemeg47rv2  34614
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