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

Theorem sbalyz 1975
 Description: Move universal quantifier in and out of substitution. Identical to sbal 1976 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 1522 . . . 4 𝑥𝑥𝜑
21nfsbxy 1916 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
3 ax-4 1488 . . . 4 (∀𝑥𝜑𝜑)
43sbimi 1738 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 → [𝑧 / 𝑦]𝜑)
52, 4alrimi 1503 . 2 ([𝑧 / 𝑦]∀𝑥𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
6 sb6 1859 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
76albii 1447 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑))
8 alcom 1455 . . . 4 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
97, 8bitri 183 . . 3 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
10 nfv 1509 . . . . . 6 𝑥 𝑦 = 𝑧
1110stdpc5 1564 . . . . 5 (∀𝑥(𝑦 = 𝑧𝜑) → (𝑦 = 𝑧 → ∀𝑥𝜑))
1211alimi 1432 . . . 4 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
13 sb2 1741 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
1412, 13syl 14 . . 3 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
159, 14sylbi 120 . 2 (∀𝑥[𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑)
165, 15impbii 125 1 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330  [wsb 1736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737 This theorem is referenced by:  sbal  1976
 Copyright terms: Public domain W3C validator