ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbco2h GIF version

Theorem sbco2h 1893
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
sbco2h.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
sbco2h ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco2h
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbco2h.1 . . . . 5 (𝜑 → ∀𝑧𝜑)
21nfi 1403 . . . 4 𝑧𝜑
32sbco2yz 1892 . . 3 ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
43sbbii 1702 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5 nfv 1473 . . 3 𝑤[𝑧 / 𝑥]𝜑
65sbco2yz 1892 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
7 nfv 1473 . . 3 𝑤𝜑
87sbco2yz 1892 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
94, 6, 83bitr3i 209 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1294  [wsb 1699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700
This theorem is referenced by:  sbco2  1894  sbco2d  1895  sbco3  1903  elsb3  1907  elsb4  1908  sb9  1910
  Copyright terms: Public domain W3C validator