ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcom2v GIF version

Theorem sbcom2v 1877
Description: Lemma for proving sbcom2 1879. It is the same as sbcom2 1879 but with additional distinct variable constraints on 𝑥 and 𝑦, and on 𝑤 and 𝑧. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2v ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Distinct variable groups:   𝑥,𝑤,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem sbcom2v
StepHypRef Expression
1 alcom 1383 . . 3 (∀𝑧𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ ∀𝑥𝑧(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)))
2 bi2.04 241 . . . . . 6 ((𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ (𝑧 = 𝑤 → (𝑥 = 𝑦𝜑)))
32albii 1375 . . . . 5 (∀𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ ∀𝑥(𝑧 = 𝑤 → (𝑥 = 𝑦𝜑)))
4 19.21v 1769 . . . . 5 (∀𝑥(𝑧 = 𝑤 → (𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)))
53, 4bitri 177 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)))
65albii 1375 . . 3 (∀𝑧𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ ∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)))
7 19.21v 1769 . . . 4 (∀𝑧(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤𝜑)))
87albii 1375 . . 3 (∀𝑥𝑧(𝑥 = 𝑦 → (𝑧 = 𝑤𝜑)) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤𝜑)))
91, 6, 83bitr3i 203 . 2 (∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤𝜑)))
10 sb6 1782 . . 3 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑤 → [𝑦 / 𝑥]𝜑))
11 sb6 1782 . . . . 5 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
1211imbi2i 219 . . . 4 ((𝑧 = 𝑤 → [𝑦 / 𝑥]𝜑) ↔ (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)))
1312albii 1375 . . 3 (∀𝑧(𝑧 = 𝑤 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)))
1410, 13bitri 177 . 2 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝜑)))
15 sb6 1782 . . 3 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑤 / 𝑧]𝜑))
16 sb6 1782 . . . . 5 ([𝑤 / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = 𝑤𝜑))
1716imbi2i 219 . . . 4 ((𝑥 = 𝑦 → [𝑤 / 𝑧]𝜑) ↔ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤𝜑)))
1817albii 1375 . . 3 (∀𝑥(𝑥 = 𝑦 → [𝑤 / 𝑧]𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤𝜑)))
1915, 18bitri 177 . 2 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤𝜑)))
209, 14, 193bitr4i 205 1 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]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-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbcom2v2  1878
  Copyright terms: Public domain W3C validator