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

Theorem csbfvg 5239
 Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
Assertion
Ref Expression
csbfvg (𝐴𝐶𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
Distinct variable group:   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem csbfvg
StepHypRef Expression
1 csbfv2g 5238 . 2 (𝐴𝐶𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴 / 𝑥𝑥))
2 csbvarg 2905 . . 3 (𝐴𝐶𝐴 / 𝑥𝑥 = 𝐴)
32fveq2d 5210 . 2 (𝐴𝐶 → (𝐹𝐴 / 𝑥𝑥) = (𝐹𝐴))
41, 3eqtrd 2088 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∈ wcel 1409  ⦋csb 2880  ‘cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator