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Theorem cdleme25cv 34458
Description: Change bound variables in cdleme25c 34455. (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 4581 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 𝑊𝑧 𝑊))
21notbid 307 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 𝑊 ↔ ¬ 𝑧 𝑊))
3 breq1 4581 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 (𝑃 𝑄) ↔ 𝑧 (𝑃 𝑄)))
43notbid 307 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 (𝑃 𝑄) ↔ ¬ 𝑧 (𝑃 𝑄)))
52, 4anbi12d 743 . . . . . . 7 (𝑠 = 𝑧 → ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ↔ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))))
6 oveq1 6534 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑠 𝑈) = (𝑧 𝑈))
7 oveq2 6535 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝑃 𝑠) = (𝑃 𝑧))
87oveq1d 6542 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑧) 𝑊))
98oveq2d 6543 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑧) 𝑊)))
106, 9oveq12d 6545 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))))
11 oveq2 6535 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑅 𝑠) = (𝑅 𝑧))
1211oveq1d 6542 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑅 𝑠) 𝑊) = ((𝑅 𝑧) 𝑊))
1310, 12oveq12d 6545 . . . . . . . . 9 (𝑠 = 𝑧 → (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
1413oveq2d 6543 . . . . . . . 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 2620 . . . . . . 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 333 . . . . . 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 3147 . . . . 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 6537 . . . . . . . . . 10 (𝐹 ((𝑅 𝑠) 𝑊)) = (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))
2120oveq2i 6538 . . . . . . . . 9 ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊))) = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2218, 21eqtri 2632 . . . . . . . 8 𝑁 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2322eqeq2i 2622 . . . . . . 7 (𝑢 = 𝑁𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))))
2423imbi2i 325 . . . . . 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 2963 . . . . 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 6537 . . . . . . . . . 10 (𝐺 ((𝑅 𝑧) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))
2928oveq2i 6538 . . . . . . . . 9 ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊))) = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3026, 29eqtri 2632 . . . . . . . 8 𝑂 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3130eqeq2i 2622 . . . . . . 7 (𝑢 = 𝑂𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))))
3231imbi2i 325 . . . . . 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 2963 . . . . 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 291 . . . 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 6506 . 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 2642 1 𝐼 = 𝐸
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4578  crio 6488  (class class class)co 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-iota 5754  df-fv 5798  df-riota 6489  df-ov 6530
This theorem is referenced by:  cdleme27a  34467
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