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

Theorem sbcnestgf 2979
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))

Proof of Theorem sbcnestgf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2842 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2 csbeq1 2936 . . . . . 6 (𝑧 = 𝐴𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 dfsbcq 2842 . . . . . 6 (𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵 → ([𝑧 / 𝑥𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
42, 3syl 14 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
51, 4bibi12d 233 . . . 4 (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
65imbi2d 228 . . 3 (𝑧 = 𝐴 → ((∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑)) ↔ (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))))
7 vex 2622 . . . . 5 𝑧 ∈ V
87a1i 9 . . . 4 (∀𝑦𝑥𝜑𝑧 ∈ V)
9 csbeq1a 2941 . . . . . 6 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
10 dfsbcq 2842 . . . . . 6 (𝐵 = 𝑧 / 𝑥𝐵 → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
119, 10syl 14 . . . . 5 (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
1211adantl 271 . . . 4 ((∀𝑦𝑥𝜑𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
13 nfnf1 1481 . . . . 5 𝑥𝑥𝜑
1413nfal 1513 . . . 4 𝑥𝑦𝑥𝜑
15 nfa1 1479 . . . . 5 𝑦𝑦𝑥𝜑
16 nfcsb1v 2963 . . . . . 6 𝑥𝑧 / 𝑥𝐵
1716a1i 9 . . . . 5 (∀𝑦𝑥𝜑𝑥𝑧 / 𝑥𝐵)
18 sp 1446 . . . . 5 (∀𝑦𝑥𝜑 → Ⅎ𝑥𝜑)
1915, 17, 18nfsbcd 2859 . . . 4 (∀𝑦𝑥𝜑 → Ⅎ𝑥[𝑧 / 𝑥𝐵 / 𝑦]𝜑)
208, 12, 14, 19sbciedf 2874 . . 3 (∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
216, 20vtoclg 2679 . 2 (𝐴𝑉 → (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
2221imp 122 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215  Vcvv 2619  [wsbc 2840  csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  csbnestgf  2980  sbcnestg  2981
  Copyright terms: Public domain W3C validator