New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cbv2h GIF version

Theorem cbv2h 1980
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cbv2h.1 (φ → (ψyψ))
cbv2h.2 (φ → (χxχ))
cbv2h.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbv2h (xyφ → (xψyχ))

Proof of Theorem cbv2h
StepHypRef Expression
1 cbv2h.1 . . 3 (φ → (ψyψ))
2 cbv2h.2 . . 3 (φ → (χxχ))
3 cbv2h.3 . . . 4 (φ → (x = y → (ψχ)))
4 bi1 178 . . . 4 ((ψχ) → (ψχ))
53, 4syl6 29 . . 3 (φ → (x = y → (ψχ)))
61, 2, 5cbv1h 1978 . 2 (xyφ → (xψyχ))
7 equcomi 1679 . . . . 5 (y = xx = y)
8 bi2 189 . . . . 5 ((ψχ) → (χψ))
97, 3, 8syl56 30 . . . 4 (φ → (y = x → (χψ)))
102, 1, 9cbv1h 1978 . . 3 (yxφ → (yχxψ))
1110a7s 1735 . 2 (xyφ → (yχxψ))
126, 11impbid 183 1 (xyφ → (xψyχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem is referenced by:  cbv2  1981  eujustALT  2207
 Copyright terms: Public domain W3C validator