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Theorem cdleme25cv 36887
 Description: Change bound variables in cdleme25c 36884. (Contributed by NM, 2-Feb-2013.)
Hypotheses
Ref Expression
cdleme25cv.f 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme25cv.n 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))
cdleme25cv.g 𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme25cv.o 𝑂 = ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊)))
cdleme25cv.i 𝐼 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁))
cdleme25cv.e 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
Assertion
Ref Expression
cdleme25cv 𝐼 = 𝐸
Distinct variable groups:   𝑧,𝑠,𝐴   ,𝑠,𝑧   ,𝑠,𝑧   ,𝑠,𝑧   𝑃,𝑠,𝑧   𝑄,𝑠,𝑧   𝑅,𝑠,𝑧   𝑈,𝑠,𝑧   𝑊,𝑠,𝑧   𝑢,𝑠,𝑧
Allowed substitution hints:   𝐴(𝑢)   𝐵(𝑧,𝑢,𝑠)   𝑃(𝑢)   𝑄(𝑢)   𝑅(𝑢)   𝑈(𝑢)   𝐸(𝑧,𝑢,𝑠)   𝐹(𝑧,𝑢,𝑠)   𝐺(𝑧,𝑢,𝑠)   𝐼(𝑧,𝑢,𝑠)   (𝑢)   (𝑢)   (𝑢)   𝑁(𝑧,𝑢,𝑠)   𝑂(𝑧,𝑢,𝑠)   𝑊(𝑢)

Proof of Theorem cdleme25cv
StepHypRef Expression
1 breq1 4926 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 𝑊𝑧 𝑊))
21notbid 310 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 𝑊 ↔ ¬ 𝑧 𝑊))
3 breq1 4926 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 (𝑃 𝑄) ↔ 𝑧 (𝑃 𝑄)))
43notbid 310 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 (𝑃 𝑄) ↔ ¬ 𝑧 (𝑃 𝑄)))
52, 4anbi12d 621 . . . . . . 7 (𝑠 = 𝑧 → ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ↔ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))))
6 oveq1 6977 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑠 𝑈) = (𝑧 𝑈))
7 oveq2 6978 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝑃 𝑠) = (𝑃 𝑧))
87oveq1d 6985 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑧) 𝑊))
98oveq2d 6986 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑧) 𝑊)))
106, 9oveq12d 6988 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))))
11 oveq2 6978 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑅 𝑠) = (𝑅 𝑧))
1211oveq1d 6985 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑅 𝑠) 𝑊) = ((𝑅 𝑧) 𝑊))
1310, 12oveq12d 6988 . . . . . . . . 9 (𝑠 = 𝑧 → (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
1413oveq2d 6986 . . . . . . . 8 (𝑠 = 𝑧 → ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))) = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))))
1514eqeq2d 2782 . . . . . . 7 (𝑠 = 𝑧 → (𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))) ↔ 𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))))
165, 15imbi12d 337 . . . . . 6 (𝑠 = 𝑧 → (((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))) ↔ ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))))))
1716cbvralv 3377 . . . . 5 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))))
18 cdleme25cv.n . . . . . . . . 9 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))
19 cdleme25cv.f . . . . . . . . . . 11 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
2019oveq1i 6980 . . . . . . . . . 10 (𝐹 ((𝑅 𝑠) 𝑊)) = (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))
2120oveq2i 6981 . . . . . . . . 9 ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊))) = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2218, 21eqtri 2796 . . . . . . . 8 𝑁 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2322eqeq2i 2784 . . . . . . 7 (𝑢 = 𝑁𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))))
2423imbi2i 328 . . . . . 6 (((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))))
2524ralbii 3109 . . . . 5 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))))
26 cdleme25cv.o . . . . . . . . 9 𝑂 = ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊)))
27 cdleme25cv.g . . . . . . . . . . 11 𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
2827oveq1i 6980 . . . . . . . . . 10 (𝐺 ((𝑅 𝑧) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))
2928oveq2i 6981 . . . . . . . . 9 ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊))) = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3026, 29eqtri 2796 . . . . . . . 8 𝑂 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3130eqeq2i 2784 . . . . . . 7 (𝑢 = 𝑂𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))))
3231imbi2i 328 . . . . . 6 (((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂) ↔ ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))))
3332ralbii 3109 . . . . 5 (∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))))
3417, 25, 333bitr4i 295 . . . 4 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
3534a1i 11 . . 3 (𝑢𝐵 → (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)))
3635riotabiia 6948 . 2 (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁)) = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
37 cdleme25cv.i . 2 𝐼 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁))
38 cdleme25cv.e . 2 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
3936, 37, 383eqtr4i 2806 1 𝐼 = 𝐸
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082   class class class wbr 4923  ℩crio 6930  (class class class)co 6970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-iota 6146  df-fv 6190  df-riota 6931  df-ov 6973 This theorem is referenced by:  cdleme27a  36896
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