Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  brabsb Structured version   GIF version

Theorem brabsb 3968
 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 3735 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
2 brabsb.1 . . 3 𝑅 = {⟨x, y⟩ ∣ φ}
32eleq2i 2082 . 2 (⟨A, B 𝑅 ↔ ⟨A, B {⟨x, y⟩ ∣ φ})
4 opelopabsb 3967 . 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 1226   ∈ wcel 1370  [wsbc 2737  ⟨cop 3349   class class class wbr 3734  {copab 3787 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789 This theorem is referenced by:  eqerlem  6044
 Copyright terms: Public domain W3C validator