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

Theorem nfsb 1958
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
nfsb.1 𝑧𝜑
Assertion
Ref Expression
nfsb 𝑧[𝑦 / 𝑥]𝜑
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfsb.1 . . . 4 𝑧𝜑
21nfsbxy 1954 . . 3 𝑧[𝑤 / 𝑥]𝜑
32nfsbxy 1954 . 2 𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4 ax-17 1537 . . . 4 (𝜑 → ∀𝑤𝜑)
54sbco2vh 1957 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
65nfbii 1484 . 2 (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
73, 6mpbi 145 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1471  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  hbsb  1961  sbco2yz  1975  sbcomxyyz  1984  hbsbd  1994  nfsb4or  2033  sb8eu  2051  nfeu  2057  cbvab  2313  cbvralf  2710  cbvrexf  2711  cbvreu  2716  cbvralsv  2734  cbvrexsv  2735  cbvrab  2750  cbvreucsf  3136  cbvrabcsf  3137  cbvopab1  4091  cbvmptf  4112  cbvmpt  4113  ralxpf  4791  rexxpf  4792  cbviota  5201  sb8iota  5203  cbvriota  5862  dfoprab4f  6218
  Copyright terms: Public domain W3C validator