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

Theorem sbalyz 1891
Description: Move universal quantifier in and out of substitution. Identical to sbal 1892 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbalyz ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbalyz
StepHypRef Expression
1 nfa1 1450 . . . 4 𝑥𝑥𝜑
21nfsbxy 1834 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
3 ax-4 1416 . . . 4 (∀𝑥𝜑𝜑)
43sbimi 1663 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 → [𝑧 / 𝑦]𝜑)
52, 4alrimi 1431 . 2 ([𝑧 / 𝑦]∀𝑥𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
6 sb6 1782 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
76albii 1375 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑))
8 alcom 1383 . . . 4 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
97, 8bitri 177 . . 3 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
10 nfv 1437 . . . . . 6 𝑥 𝑦 = 𝑧
1110stdpc5 1492 . . . . 5 (∀𝑥(𝑦 = 𝑧𝜑) → (𝑦 = 𝑧 → ∀𝑥𝜑))
1211alimi 1360 . . . 4 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
13 sb2 1666 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
1412, 13syl 14 . . 3 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
159, 14sylbi 118 . 2 (∀𝑥[𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑)
165, 15impbii 121 1 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  [wsb 1661
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
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sbal  1892
  Copyright terms: Public domain W3C validator