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Theorem recexprlemrnd 6600
 Description: B is rounded. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
Assertion
Ref Expression
recexprlemrnd (A P → (𝑞 Q (𝑞 (1stB) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))) 𝑟 Q (𝑟 (2ndB) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))))
Distinct variable groups:   𝑟,𝑞,x,y,A   B,𝑞,𝑟,x,y

Proof of Theorem recexprlemrnd
StepHypRef Expression
1 recexpr.1 . . . . . 6 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
21recexprlemopl 6596 . . . . 5 ((A P 𝑞 Q 𝑞 (1stB)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))
323expia 1105 . . . 4 ((A P 𝑞 Q) → (𝑞 (1stB) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))))
41recexprlemlol 6597 . . . 4 ((A P 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)) → 𝑞 (1stB)))
53, 4impbid 120 . . 3 ((A P 𝑞 Q) → (𝑞 (1stB) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))))
65ralrimiva 2386 . 2 (A P𝑞 Q (𝑞 (1stB) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))))
71recexprlemopu 6598 . . . . 5 ((A P 𝑟 Q 𝑟 (2ndB)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))
873expia 1105 . . . 4 ((A P 𝑟 Q) → (𝑟 (2ndB) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB))))
91recexprlemupu 6599 . . . 4 ((A P 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)) → 𝑟 (2ndB)))
108, 9impbid 120 . . 3 ((A P 𝑟 Q) → (𝑟 (2ndB) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB))))
1110ralrimiva 2386 . 2 (A P𝑟 Q (𝑟 (2ndB) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB))))
126, 11jca 290 1 (A P → (𝑞 Q (𝑞 (1stB) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))) 𝑟 Q (𝑟 (2ndB) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  *Qcrq 6268
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