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Theorem opabbi2dv 4428
 Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2153. (Contributed by NM, 24-Feb-2014.)
Hypotheses
Ref Expression
opabbi2dv.1 Rel A
opabbi2dv.3 (φ → (⟨x, y Aψ))
Assertion
Ref Expression
opabbi2dv (φA = {⟨x, y⟩ ∣ ψ})
Distinct variable groups:   x,y,A   φ,x,y
Allowed substitution hints:   ψ(x,y)

Proof of Theorem opabbi2dv
StepHypRef Expression
1 opabbi2dv.1 . . 3 Rel A
2 opabid2 4410 . . 3 (Rel A → {⟨x, y⟩ ∣ ⟨x, y A} = A)
31, 2ax-mp 7 . 2 {⟨x, y⟩ ∣ ⟨x, y A} = A
4 opabbi2dv.3 . . 3 (φ → (⟨x, y Aψ))
54opabbidv 3814 . 2 (φ → {⟨x, y⟩ ∣ ⟨x, y A} = {⟨x, y⟩ ∣ ψ})
63, 5syl5eqr 2083 1 (φA = {⟨x, y⟩ ∣ ψ})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ⟨cop 3370  {copab 3808  Rel wrel 4293 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-ral 2305  df-rex 2306  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  df-xp 4294  df-rel 4295 This theorem is referenced by: (None)
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