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Theorem genpdflem 7279
Description: Simplification of upper or lower cut expression. Lemma for genpdf 7280. (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 114 . . . . . . . 8 (𝜑 → (𝑟𝐴𝑟Q))
32pm4.71rd 389 . . . . . . 7 (𝜑 → (𝑟𝐴 ↔ (𝑟Q𝑟𝐴)))
43anbi1d 458 . . . . . 6 (𝜑 → ((𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
54exbidv 1779 . . . . 5 (𝜑 → (∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
6 3anass 949 . . . . . . . . . 10 ((𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
76rexbii 2417 . . . . . . . . 9 (∃𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠Q (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
8 r19.42v 2563 . . . . . . . . 9 (∃𝑠Q (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
97, 8bitri 183 . . . . . . . 8 (∃𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
109rexbii 2417 . . . . . . 7 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟Q (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
11 df-rex 2397 . . . . . . 7 (∃𝑟Q (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1210, 11bitri 183 . . . . . 6 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
13 anass 396 . . . . . . 7 (((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1413exbii 1567 . . . . . 6 (∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1512, 14bitr4i 186 . . . . 5 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
165, 15syl6rbbr 198 . . . 4 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
17 df-rex 2397 . . . 4 (∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
1816, 17syl6bbr 197 . . 3 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
19 genpdflem.s . . . . . . . . . 10 ((𝜑𝑠𝐵) → 𝑠Q)
2019ex 114 . . . . . . . . 9 (𝜑 → (𝑠𝐵𝑠Q))
2120pm4.71rd 389 . . . . . . . 8 (𝜑 → (𝑠𝐵 ↔ (𝑠Q𝑠𝐵)))
2221anbi1d 458 . . . . . . 7 (𝜑 → ((𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠))))
2322exbidv 1779 . . . . . 6 (𝜑 → (∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠))))
24 df-rex 2397 . . . . . . 7 (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
25 anass 396 . . . . . . . 8 (((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)) ↔ (𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
2625exbii 1567 . . . . . . 7 (∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
2724, 26bitr4i 186 . . . . . 6 (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)))
2823, 27syl6rbbr 198 . . . . 5 (𝜑 → (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
29 df-rex 2397 . . . . 5 (∃𝑠𝐵 𝑞 = (𝑟𝐺𝑠) ↔ ∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠)))
3028, 29syl6bbr 197 . . . 4 (𝜑 → (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3130rexbidv 2413 . . 3 (𝜑 → (∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3218, 31bitrd 187 . 2 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3332rabbidv 2647 1 (𝜑 → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945   = wceq 1314  wex 1451  wcel 1463  wrex 2392  {crab 2395  (class class class)co 5740  Qcnq 7052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-ral 2396  df-rex 2397  df-rab 2400
This theorem is referenced by:  genpdf  7280
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