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Theorem opelopab2a 3993
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1 ((x = A y = B) → (φψ))
Assertion
Ref Expression
opelopab2a ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ ψ))
Distinct variable groups:   x,y,A   x,B,y   ψ,x,y   x,𝐶,y   x,𝐷,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem opelopab2a
StepHypRef Expression
1 eleq1 2097 . . . . 5 (x = A → (x 𝐶A 𝐶))
2 eleq1 2097 . . . . 5 (y = B → (y 𝐷B 𝐷))
31, 2bi2anan9 538 . . . 4 ((x = A y = B) → ((x 𝐶 y 𝐷) ↔ (A 𝐶 B 𝐷)))
4 opelopabga.1 . . . 4 ((x = A y = B) → (φψ))
53, 4anbi12d 442 . . 3 ((x = A y = B) → (((x 𝐶 y 𝐷) φ) ↔ ((A 𝐶 B 𝐷) ψ)))
65opelopabga 3991 . 2 ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ ((A 𝐶 B 𝐷) ψ)))
76bianabs 543 1 ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  cop 3370  {copab 3808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by:  opelopab2  3998  brab2a  4336  brab2ga  4358  ltdfpr  6488
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