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Theorem brabsb 3989
Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
Hypothesis
Ref Expression
brabsb.1 𝑅 = {⟨x, y⟩ ∣ φ}
Assertion
Ref Expression
brabsb (A𝑅B[A / x][B / y]φ)
Distinct variable groups:   x,y   x,B
Allowed substitution hints:   φ(x,y)   A(x,y)   B(y)   𝑅(x,y)

Proof of Theorem brabsb
StepHypRef Expression
1 df-br 3756 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
2 brabsb.1 . . 3 𝑅 = {⟨x, y⟩ ∣ φ}
32eleq2i 2101 . 2 (⟨A, B 𝑅 ↔ ⟨A, B {⟨x, y⟩ ∣ φ})
4 opelopabsb 3988 . 2 (⟨A, B {⟨x, y⟩ ∣ φ} ↔ [A / x][B / y]φ)
51, 3, 43bitri 195 1 (A𝑅B[A / x][B / y]φ)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  [wsbc 2758  cop 3370   class class class wbr 3755  {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-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810
This theorem is referenced by:  eqerlem  6073
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