New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  2sb6 GIF version

Theorem 2sb6 2113
 Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb6 ([z / x][w / y]φxy((x = z y = w) → φ))
Distinct variable groups:   x,y,z   y,w
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem 2sb6
StepHypRef Expression
1 sb6 2099 . 2 ([z / x][w / y]φx(x = z → [w / y]φ))
2 19.21v 1890 . . . 4 (y(x = z → (y = wφ)) ↔ (x = zy(y = wφ)))
3 impexp 433 . . . . 5 (((x = z y = w) → φ) ↔ (x = z → (y = wφ)))
43albii 1566 . . . 4 (y((x = z y = w) → φ) ↔ y(x = z → (y = wφ)))
5 sb6 2099 . . . . 5 ([w / y]φy(y = wφ))
65imbi2i 303 . . . 4 ((x = z → [w / y]φ) ↔ (x = zy(y = wφ)))
72, 4, 63bitr4ri 269 . . 3 ((x = z → [w / y]φ) ↔ y((x = z y = w) → φ))
87albii 1566 . 2 (x(x = z → [w / y]φ) ↔ xy((x = z y = w) → φ))
91, 8bitri 240 1 ([z / x][w / y]φxy((x = z y = w) → φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  2eu6  2289
 Copyright terms: Public domain W3C validator