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Theorem cdleme31se2 36191
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 2902 . . . . 5 𝑡(𝑃 𝑄)
2 nfcv 2902 . . . . 5 𝑡
3 nfcsb1v 3690 . . . . . 6 𝑡𝑆 / 𝑡𝐷
4 nfcv 2902 . . . . . 6 𝑡
5 nfcv 2902 . . . . . 6 𝑡((𝑅 𝑆) 𝑊)
63, 4, 5nfov 6840 . . . . 5 𝑡(𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))
71, 2, 6nfov 6840 . . . 4 𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
87a1i 11 . . 3 (𝑆𝐴𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
9 csbeq1a 3683 . . . . 5 (𝑡 = 𝑆𝐷 = 𝑆 / 𝑡𝐷)
10 oveq2 6822 . . . . . 6 (𝑡 = 𝑆 → (𝑅 𝑡) = (𝑅 𝑆))
1110oveq1d 6829 . . . . 5 (𝑡 = 𝑆 → ((𝑅 𝑡) 𝑊) = ((𝑅 𝑆) 𝑊))
129, 11oveq12d 6832 . . . 4 (𝑡 = 𝑆 → (𝐷 ((𝑅 𝑡) 𝑊)) = (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
1312oveq2d 6830 . . 3 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
148, 13csbiegf 3698 . 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 4136 . 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 2819 1 (𝑆𝐴𝑆 / 𝑡𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wnfc 2889  csb 3674  (class class class)co 6814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817
This theorem is referenced by:  cdlemeg47rv2  36318
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