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Theorem cdleme25cv 37376
Description: Change bound variables in cdleme25c 37373. (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 5061 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 𝑊𝑧 𝑊))
21notbid 319 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 𝑊 ↔ ¬ 𝑧 𝑊))
3 breq1 5061 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 (𝑃 𝑄) ↔ 𝑧 (𝑃 𝑄)))
43notbid 319 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 (𝑃 𝑄) ↔ ¬ 𝑧 (𝑃 𝑄)))
52, 4anbi12d 630 . . . . . . 7 (𝑠 = 𝑧 → ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ↔ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))))
6 oveq1 7152 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑠 𝑈) = (𝑧 𝑈))
7 oveq2 7153 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝑃 𝑠) = (𝑃 𝑧))
87oveq1d 7160 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑧) 𝑊))
98oveq2d 7161 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑧) 𝑊)))
106, 9oveq12d 7163 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))))
11 oveq2 7153 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑅 𝑠) = (𝑅 𝑧))
1211oveq1d 7160 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑅 𝑠) 𝑊) = ((𝑅 𝑧) 𝑊))
1310, 12oveq12d 7163 . . . . . . . . 9 (𝑠 = 𝑧 → (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
1413oveq2d 7161 . . . . . . . 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 2832 . . . . . . 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 346 . . . . . 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))))))
1716cbvralvw 3450 . . . . 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 7155 . . . . . . . . . 10 (𝐹 ((𝑅 𝑠) 𝑊)) = (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))
2120oveq2i 7156 . . . . . . . . 9 ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊))) = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2218, 21eqtri 2844 . . . . . . . 8 𝑁 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2322eqeq2i 2834 . . . . . . 7 (𝑢 = 𝑁𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))))
2423imbi2i 337 . . . . . 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 3165 . . . . 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 7155 . . . . . . . . . 10 (𝐺 ((𝑅 𝑧) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))
2928oveq2i 7156 . . . . . . . . 9 ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊))) = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3026, 29eqtri 2844 . . . . . . . 8 𝑂 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3130eqeq2i 2834 . . . . . . 7 (𝑢 = 𝑂𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))))
3231imbi2i 337 . . . . . 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 3165 . . . . 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 304 . . . 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 7123 . 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 2854 1 𝐼 = 𝐸
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3138   class class class wbr 5058  crio 7102  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-iota 6308  df-fv 6357  df-riota 7103  df-ov 7148
This theorem is referenced by:  cdleme27a  37385
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