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Theorem ltdfpr 6489
 Description: More convenient form of df-iltp 6453. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltdfpr ((A P B P) → (A<P B𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
Distinct variable groups:   A,𝑞   B,𝑞

Proof of Theorem ltdfpr
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3756 . . 3 (A<P B ↔ ⟨A, B <P )
2 df-iltp 6453 . . . 4 <P = {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))}
32eleq2i 2101 . . 3 (⟨A, B <P ↔ ⟨A, B {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))})
41, 3bitri 173 . 2 (A<P B ↔ ⟨A, B {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))})
5 simpl 102 . . . . . . 7 ((x = A y = B) → x = A)
65fveq2d 5125 . . . . . 6 ((x = A y = B) → (2ndx) = (2ndA))
76eleq2d 2104 . . . . 5 ((x = A y = B) → (𝑞 (2ndx) ↔ 𝑞 (2ndA)))
8 simpr 103 . . . . . . 7 ((x = A y = B) → y = B)
98fveq2d 5125 . . . . . 6 ((x = A y = B) → (1sty) = (1stB))
109eleq2d 2104 . . . . 5 ((x = A y = B) → (𝑞 (1sty) ↔ 𝑞 (1stB)))
117, 10anbi12d 442 . . . 4 ((x = A y = B) → ((𝑞 (2ndx) 𝑞 (1sty)) ↔ (𝑞 (2ndA) 𝑞 (1stB))))
1211rexbidv 2321 . . 3 ((x = A y = B) → (𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)) ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
1312opelopab2a 3993 . 2 ((A P B P) → (⟨A, B {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))} ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
144, 13syl5bb 181 1 ((A P B P) → (A<P B𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  {copab 3808  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275
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