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Theorem inopab 4411
Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
inopab ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ψ)}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem inopab
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4407 . . 3 Rel {⟨x, y⟩ ∣ φ}
2 relin1 4398 . . 3 (Rel {⟨x, y⟩ ∣ φ} → Rel ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}))
31, 2ax-mp 7 . 2 Rel ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ})
4 relopab 4407 . 2 Rel {⟨x, y⟩ ∣ (φ ψ)}
5 sban 1826 . . . 4 ([w / y]([z / x]φ [z / x]ψ) ↔ ([w / y][z / x]φ [w / y][z / x]ψ))
6 sban 1826 . . . . 5 ([z / x](φ ψ) ↔ ([z / x]φ [z / x]ψ))
76sbbii 1645 . . . 4 ([w / y][z / x](φ ψ) ↔ [w / y]([z / x]φ [z / x]ψ))
8 opelopabsbALT 3987 . . . . 5 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ [w / y][z / x]φ)
9 opelopabsbALT 3987 . . . . 5 (⟨z, w {⟨x, y⟩ ∣ ψ} ↔ [w / y][z / x]ψ)
108, 9anbi12i 433 . . . 4 ((⟨z, w {⟨x, y⟩ ∣ φ} z, w {⟨x, y⟩ ∣ ψ}) ↔ ([w / y][z / x]φ [w / y][z / x]ψ))
115, 7, 103bitr4ri 202 . . 3 ((⟨z, w {⟨x, y⟩ ∣ φ} z, w {⟨x, y⟩ ∣ ψ}) ↔ [w / y][z / x](φ ψ))
12 elin 3120 . . 3 (⟨z, w ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ (⟨z, w {⟨x, y⟩ ∣ φ} z, w {⟨x, y⟩ ∣ ψ}))
13 opelopabsbALT 3987 . . 3 (⟨z, w {⟨x, y⟩ ∣ (φ ψ)} ↔ [w / y][z / x](φ ψ))
1411, 12, 133bitr4i 201 . 2 (⟨z, w ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ ⟨z, w {⟨x, y⟩ ∣ (φ ψ)})
153, 4, 14eqrelriiv 4377 1 ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ψ)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  [wsb 1642  cin 2910  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-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:  inxp  4413  resopab  4595  cnvin  4674  fndmin  5217  enq0enq  6414
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