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Theorem cdleme31se2 36404
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 2941 . . . . 5 𝑡(𝑃 𝑄)
2 nfcv 2941 . . . . 5 𝑡
3 nfcsb1v 3744 . . . . . 6 𝑡𝑆 / 𝑡𝐷
4 nfcv 2941 . . . . . 6 𝑡
5 nfcv 2941 . . . . . 6 𝑡((𝑅 𝑆) 𝑊)
63, 4, 5nfov 6908 . . . . 5 𝑡(𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))
71, 2, 6nfov 6908 . . . 4 𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
87a1i 11 . . 3 (𝑆𝐴𝑡((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
9 csbeq1a 3737 . . . . 5 (𝑡 = 𝑆𝐷 = 𝑆 / 𝑡𝐷)
10 oveq2 6886 . . . . . 6 (𝑡 = 𝑆 → (𝑅 𝑡) = (𝑅 𝑆))
1110oveq1d 6893 . . . . 5 (𝑡 = 𝑆 → ((𝑅 𝑡) 𝑊) = ((𝑅 𝑆) 𝑊))
129, 11oveq12d 6896 . . . 4 (𝑡 = 𝑆 → (𝐷 ((𝑅 𝑡) 𝑊)) = (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊)))
1312oveq2d 6894 . . 3 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑆 / 𝑡𝐷 ((𝑅 𝑆) 𝑊))))
148, 13csbiegf 3752 . 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 4188 . 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 2858 1 (𝑆𝐴𝑆 / 𝑡𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wnfc 2928  csb 3728  (class class class)co 6878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881
This theorem is referenced by:  cdlemeg47rv2  36531
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