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

Theorem sbbi 1830
 Description: Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbbi ([y / x](φψ) ↔ ([y / x]φ ↔ [y / x]ψ))

Proof of Theorem sbbi
StepHypRef Expression
1 dfbi2 368 . . 3 ((φψ) ↔ ((φψ) (ψφ)))
21sbbii 1645 . 2 ([y / x](φψ) ↔ [y / x]((φψ) (ψφ)))
3 sbim 1824 . . . 4 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
4 sbim 1824 . . . 4 ([y / x](ψφ) ↔ ([y / x]ψ → [y / x]φ))
53, 4anbi12i 433 . . 3 (([y / x](φψ) [y / x](ψφ)) ↔ (([y / x]φ → [y / x]ψ) ([y / x]ψ → [y / x]φ)))
6 sban 1826 . . 3 ([y / x]((φψ) (ψφ)) ↔ ([y / x](φψ) [y / x](ψφ)))
7 dfbi2 368 . . 3 (([y / x]φ ↔ [y / x]ψ) ↔ (([y / x]φ → [y / x]ψ) ([y / x]ψ → [y / x]φ)))
85, 6, 73bitr4i 201 . 2 ([y / x]((φψ) (ψφ)) ↔ ([y / x]φ ↔ [y / x]ψ))
92, 8bitri 173 1 ([y / x](φψ) ↔ ([y / x]φ ↔ [y / x]ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  sblbis  1831  sbrbis  1832  sbco  1839  sbcocom  1841  elsb3  1849  elsb4  1850  sb8eu  1910  sb8euh  1920  pm13.183  2675  sbcbig  2803  sb8iota  4817
 Copyright terms: Public domain W3C validator