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Theorem genpdflem 6489
Description: Simplification of upper or lower cut expression. Lemma for genpdf 6490. (Contributed by Jim Kingdon, 30-Sep-2019.)
Hypotheses
Ref Expression
genpdflem.r ((φ 𝑟 A) → 𝑟 Q)
genpdflem.s ((φ 𝑠 B) → 𝑠 Q)
Assertion
Ref Expression
genpdflem (φ → {𝑞 Q𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠))} = {𝑞 Q𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)})
Distinct variable groups:   A,𝑠   φ,𝑞,𝑟,𝑠
Allowed substitution hints:   A(𝑟,𝑞)   B(𝑠,𝑟,𝑞)   𝐺(𝑠,𝑟,𝑞)

Proof of Theorem genpdflem
StepHypRef Expression
1 genpdflem.r . . . . . . . . 9 ((φ 𝑟 A) → 𝑟 Q)
21ex 108 . . . . . . . 8 (φ → (𝑟 A𝑟 Q))
32pm4.71rd 374 . . . . . . 7 (φ → (𝑟 A ↔ (𝑟 Q 𝑟 A)))
43anbi1d 438 . . . . . 6 (φ → ((𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))) ↔ ((𝑟 Q 𝑟 A) 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
54exbidv 1703 . . . . 5 (φ → (𝑟(𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))) ↔ 𝑟((𝑟 Q 𝑟 A) 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
6 3anass 888 . . . . . . . . . 10 ((𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟 A (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
76rexbii 2325 . . . . . . . . 9 (𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠 Q (𝑟 A (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
8 r19.42v 2461 . . . . . . . . 9 (𝑠 Q (𝑟 A (𝑠 B 𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
97, 8bitri 173 . . . . . . . 8 (𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
109rexbii 2325 . . . . . . 7 (𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟 Q (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
11 df-rex 2306 . . . . . . 7 (𝑟 Q (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))) ↔ 𝑟(𝑟 Q (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
1210, 11bitri 173 . . . . . 6 (𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟(𝑟 Q (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
13 anass 381 . . . . . . 7 (((𝑟 Q 𝑟 A) 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟 Q (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
1413exbii 1493 . . . . . 6 (𝑟((𝑟 Q 𝑟 A) 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))) ↔ 𝑟(𝑟 Q (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
1512, 14bitr4i 176 . . . . 5 (𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟((𝑟 Q 𝑟 A) 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
165, 15syl6rbbr 188 . . . 4 (φ → (𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟(𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)))))
17 df-rex 2306 . . . 4 (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟(𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
1816, 17syl6bbr 187 . . 3 (φ → (𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
19 genpdflem.s . . . . . . . . . 10 ((φ 𝑠 B) → 𝑠 Q)
2019ex 108 . . . . . . . . 9 (φ → (𝑠 B𝑠 Q))
2120pm4.71rd 374 . . . . . . . 8 (φ → (𝑠 B ↔ (𝑠 Q 𝑠 B)))
2221anbi1d 438 . . . . . . 7 (φ → ((𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ ((𝑠 Q 𝑠 B) 𝑞 = (𝑟𝐺𝑠))))
2322exbidv 1703 . . . . . 6 (φ → (𝑠(𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠((𝑠 Q 𝑠 B) 𝑞 = (𝑟𝐺𝑠))))
24 df-rex 2306 . . . . . . 7 (𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠(𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
25 anass 381 . . . . . . . 8 (((𝑠 Q 𝑠 B) 𝑞 = (𝑟𝐺𝑠)) ↔ (𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
2625exbii 1493 . . . . . . 7 (𝑠((𝑠 Q 𝑠 B) 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠(𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠))))
2724, 26bitr4i 176 . . . . . 6 (𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠((𝑠 Q 𝑠 B) 𝑞 = (𝑟𝐺𝑠)))
2823, 27syl6rbbr 188 . . . . 5 (φ → (𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠(𝑠 B 𝑞 = (𝑟𝐺𝑠))))
29 df-rex 2306 . . . . 5 (𝑠 B 𝑞 = (𝑟𝐺𝑠) ↔ 𝑠(𝑠 B 𝑞 = (𝑟𝐺𝑠)))
3028, 29syl6bbr 187 . . . 4 (φ → (𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑠 B 𝑞 = (𝑟𝐺𝑠)))
3130rexbidv 2321 . . 3 (φ → (𝑟 A 𝑠 Q (𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)))
3218, 31bitrd 177 . 2 (φ → (𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)) ↔ 𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)))
3332rabbidv 2543 1 (φ → {𝑞 Q𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠))} = {𝑞 Q𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  (class class class)co 5455  Qcnq 6264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-ral 2305  df-rex 2306  df-rab 2309
This theorem is referenced by:  genpdf  6490
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