ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbi2v GIF version

Theorem sbi2v 1848
Description: Reverse direction of sbimv 1849. (Contributed by Jim Kingdon, 18-Jan-2018.)
Assertion
Ref Expression
sbi2v (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbi2v
StepHypRef Expression
1 19.38 1639 . . 3 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)) → ∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
2 pm3.3 259 . . . . 5 (((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦𝜓))))
3 pm2.04 82 . . . . 5 ((𝜑 → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑𝜓)))
42, 3syli 37 . . . 4 (((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑𝜓)))
54alimi 1416 . . 3 (∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
61, 5syl 14 . 2 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
7 sb5 1843 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 sb6 1842 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
97, 8imbi12i 238 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
10 sb6 1842 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
116, 9, 103imtr4i 200 1 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314  wex 1453  [wsb 1720
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-sb 1721
This theorem is referenced by:  sbimv  1849
  Copyright terms: Public domain W3C validator