ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb6rf GIF version

Theorem sb6rf 1853
Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb5rf.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb6rf (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Proof of Theorem sb6rf
StepHypRef Expression
1 sb5rf.1 . . 3 (𝜑 → ∀𝑦𝜑)
2 sbequ1 1768 . . . . 5 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
32equcoms 1708 . . . 4 (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
43com12 30 . . 3 (𝜑 → (𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
51, 4alrimih 1469 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6 sb2 1767 . . 3 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
71sbid2h 1849 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
86, 7sylib 122 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → 𝜑)
95, 8impbii 126 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  [wsb 1762
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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  2sb6rf  1990  eu1  2051
  Copyright terms: Public domain W3C validator