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Theorem cdleme25cv 37653
Description: Change bound variables in cdleme25c 37650. (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 5036 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 𝑊𝑧 𝑊))
21notbid 321 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 𝑊 ↔ ¬ 𝑧 𝑊))
3 breq1 5036 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑠 (𝑃 𝑄) ↔ 𝑧 (𝑃 𝑄)))
43notbid 321 . . . . . . . 8 (𝑠 = 𝑧 → (¬ 𝑠 (𝑃 𝑄) ↔ ¬ 𝑧 (𝑃 𝑄)))
52, 4anbi12d 633 . . . . . . 7 (𝑠 = 𝑧 → ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ↔ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))))
6 oveq1 7146 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑠 𝑈) = (𝑧 𝑈))
7 oveq2 7147 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝑃 𝑠) = (𝑃 𝑧))
87oveq1d 7154 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑧) 𝑊))
98oveq2d 7155 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑧) 𝑊)))
106, 9oveq12d 7157 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))))
11 oveq2 7147 . . . . . . . . . . 11 (𝑠 = 𝑧 → (𝑅 𝑠) = (𝑅 𝑧))
1211oveq1d 7154 . . . . . . . . . 10 (𝑠 = 𝑧 → ((𝑅 𝑠) 𝑊) = ((𝑅 𝑧) 𝑊))
1310, 12oveq12d 7157 . . . . . . . . 9 (𝑠 = 𝑧 → (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
1413oveq2d 7155 . . . . . . . 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 2812 . . . . . . 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 348 . . . . . 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 3399 . . . . 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 7149 . . . . . . . . . 10 (𝐹 ((𝑅 𝑠) 𝑊)) = (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))
2120oveq2i 7150 . . . . . . . . 9 ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊))) = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2218, 21eqtri 2824 . . . . . . . 8 𝑁 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊)))
2322eqeq2i 2814 . . . . . . 7 (𝑢 = 𝑁𝑢 = ((𝑃 𝑄) (((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) ((𝑅 𝑠) 𝑊))))
2423imbi2i 339 . . . . . 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 3136 . . . . 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 7149 . . . . . . . . . 10 (𝐺 ((𝑅 𝑧) 𝑊)) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))
2928oveq2i 7150 . . . . . . . . 9 ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊))) = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3026, 29eqtri 2824 . . . . . . . 8 𝑂 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊)))
3130eqeq2i 2814 . . . . . . 7 (𝑢 = 𝑂𝑢 = ((𝑃 𝑄) (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) ((𝑅 𝑧) 𝑊))))
3231imbi2i 339 . . . . . 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 3136 . . . . 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 306 . . . 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 7117 . 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 2834 1 𝐼 = 𝐸
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109   class class class wbr 5033  crio 7096  (class class class)co 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-riota 7097  df-ov 7142
This theorem is referenced by:  cdleme27a  37662
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