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Theorem genpdflem 7045
Description: Simplification of upper or lower cut expression. Lemma for genpdf 7046. (Contributed by Jim Kingdon, 30-Sep-2019.)
Hypotheses
Ref Expression
genpdflem.r ((𝜑𝑟𝐴) → 𝑟Q)
genpdflem.s ((𝜑𝑠𝐵) → 𝑠Q)
Assertion
Ref Expression
genpdflem (𝜑 → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟𝐴𝑠𝐵 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 ((𝜑𝑟𝐴) → 𝑟Q)
21ex 113 . . . . . . . 8 (𝜑 → (𝑟𝐴𝑟Q))
32pm4.71rd 386 . . . . . . 7 (𝜑 → (𝑟𝐴 ↔ (𝑟Q𝑟𝐴)))
43anbi1d 453 . . . . . 6 (𝜑 → ((𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
54exbidv 1753 . . . . 5 (𝜑 → (∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
6 3anass 928 . . . . . . . . . 10 ((𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
76rexbii 2385 . . . . . . . . 9 (∃𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠Q (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
8 r19.42v 2524 . . . . . . . . 9 (∃𝑠Q (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
97, 8bitri 182 . . . . . . . 8 (∃𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
109rexbii 2385 . . . . . . 7 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟Q (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
11 df-rex 2365 . . . . . . 7 (∃𝑟Q (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1210, 11bitri 182 . . . . . 6 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
13 anass 393 . . . . . . 7 (((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1413exbii 1541 . . . . . 6 (∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1512, 14bitr4i 185 . . . . 5 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
165, 15syl6rbbr 197 . . . 4 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
17 df-rex 2365 . . . 4 (∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
1816, 17syl6bbr 196 . . 3 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
19 genpdflem.s . . . . . . . . . 10 ((𝜑𝑠𝐵) → 𝑠Q)
2019ex 113 . . . . . . . . 9 (𝜑 → (𝑠𝐵𝑠Q))
2120pm4.71rd 386 . . . . . . . 8 (𝜑 → (𝑠𝐵 ↔ (𝑠Q𝑠𝐵)))
2221anbi1d 453 . . . . . . 7 (𝜑 → ((𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠))))
2322exbidv 1753 . . . . . 6 (𝜑 → (∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠))))
24 df-rex 2365 . . . . . . 7 (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
25 anass 393 . . . . . . . 8 (((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)) ↔ (𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
2625exbii 1541 . . . . . . 7 (∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
2724, 26bitr4i 185 . . . . . 6 (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)))
2823, 27syl6rbbr 197 . . . . 5 (𝜑 → (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
29 df-rex 2365 . . . . 5 (∃𝑠𝐵 𝑞 = (𝑟𝐺𝑠) ↔ ∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠)))
3028, 29syl6bbr 196 . . . 4 (𝜑 → (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3130rexbidv 2381 . . 3 (𝜑 → (∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3218, 31bitrd 186 . 2 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3332rabbidv 2608 1 (𝜑 → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wex 1426  wcel 1438  wrex 2360  {crab 2363  (class class class)co 5634  Qcnq 6818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-ral 2364  df-rex 2365  df-rab 2368
This theorem is referenced by:  genpdf  7046
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