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Theorem genpdflem 7421
 Description: Simplification of upper or lower cut expression. Lemma for genpdf 7422. (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 3anass 967 . . . . . . . . . 10 ((𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
21rexbii 2464 . . . . . . . . 9 (∃𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠Q (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
3 r19.42v 2614 . . . . . . . . 9 (∃𝑠Q (𝑟𝐴 ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
42, 3bitri 183 . . . . . . . 8 (∃𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
54rexbii 2464 . . . . . . 7 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟Q (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
6 df-rex 2441 . . . . . . 7 (∃𝑟Q (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
75, 6bitri 183 . . . . . 6 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
8 anass 399 . . . . . . 7 (((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ (𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
98exbii 1585 . . . . . 6 (∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟(𝑟Q ∧ (𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
107, 9bitr4i 186 . . . . 5 (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
11 genpdflem.r . . . . . . . . 9 ((𝜑𝑟𝐴) → 𝑟Q)
1211ex 114 . . . . . . . 8 (𝜑 → (𝑟𝐴𝑟Q))
1312pm4.71rd 392 . . . . . . 7 (𝜑 → (𝑟𝐴 ↔ (𝑟Q𝑟𝐴)))
1413anbi1d 461 . . . . . 6 (𝜑 → ((𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1514exbidv 1805 . . . . 5 (𝜑 → (∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))) ↔ ∃𝑟((𝑟Q𝑟𝐴) ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
1610, 15bitr4id 198 . . . 4 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)))))
17 df-rex 2441 . . . 4 (∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟(𝑟𝐴 ∧ ∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
1816, 17bitr4di 197 . . 3 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
19 df-rex 2441 . . . . . . 7 (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
20 anass 399 . . . . . . . 8 (((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)) ↔ (𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
2120exbii 1585 . . . . . . 7 (∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠Q ∧ (𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
2219, 21bitr4i 186 . . . . . 6 (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠)))
23 genpdflem.s . . . . . . . . . 10 ((𝜑𝑠𝐵) → 𝑠Q)
2423ex 114 . . . . . . . . 9 (𝜑 → (𝑠𝐵𝑠Q))
2524pm4.71rd 392 . . . . . . . 8 (𝜑 → (𝑠𝐵 ↔ (𝑠Q𝑠𝐵)))
2625anbi1d 461 . . . . . . 7 (𝜑 → ((𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠))))
2726exbidv 1805 . . . . . 6 (𝜑 → (∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠((𝑠Q𝑠𝐵) ∧ 𝑞 = (𝑟𝐺𝑠))))
2822, 27bitr4id 198 . . . . 5 (𝜑 → (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠))))
29 df-rex 2441 . . . . 5 (∃𝑠𝐵 𝑞 = (𝑟𝐺𝑠) ↔ ∃𝑠(𝑠𝐵𝑞 = (𝑟𝐺𝑠)))
3028, 29bitr4di 197 . . . 4 (𝜑 → (∃𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3130rexbidv 2458 . . 3 (𝜑 → (∃𝑟𝐴𝑠Q (𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3218, 31bitrd 187 . 2 (𝜑 → (∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠)) ↔ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)))
3332rabbidv 2701 1 (𝜑 → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∃wrex 2436  {crab 2439  (class class class)co 5821  Qcnq 7194 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-ral 2440  df-rex 2441  df-rab 2444 This theorem is referenced by:  genpdf  7422
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