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

Theorem cbval 1984
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
cbval.1 yφ
cbval.2 xψ
cbval.3 (x = y → (φψ))
Assertion
Ref Expression
cbval (xφyψ)

Proof of Theorem cbval
StepHypRef Expression
1 cbval.1 . . 3 yφ
2 cbval.2 . . 3 xψ
3 cbval.3 . . . 4 (x = y → (φψ))
43biimpd 198 . . 3 (x = y → (φψ))
51, 2, 4cbv3 1982 . 2 (xφyψ)
63biimprd 214 . . . 4 (x = y → (ψφ))
76equcoms 1681 . . 3 (y = x → (ψφ))
82, 1, 7cbv3 1982 . 2 (yψxφ)
95, 8impbii 180 1 (xφyψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544 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:  cbvex  1985  cbvalv  2002  cbval2  2004  sb8eu  2222  abbi  2463  cleqf  2513  cbvralf  2829  ralab2  3001  cbvralcsf  3198  dfss2f  3264  elintab  3937  cbviota  4344  sb8iota  4346  dffun6f  5123
 Copyright terms: Public domain W3C validator