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Theorem cbvexd 1799
 Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
Hypotheses
Ref Expression
cbvexd.1 yφ
cbvexd.2 (φ → Ⅎyψ)
cbvexd.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbvexd (φ → (xψyχ))
Distinct variable groups:   φ,x   χ,x
Allowed substitution hints:   φ(y)   ψ(x,y)   χ(y)

Proof of Theorem cbvexd
StepHypRef Expression
1 cbvexd.1 . . 3 yφ
21nfri 1409 . 2 (φyφ)
3 cbvexd.2 . . 3 (φ → Ⅎyψ)
43nfrd 1410 . 2 (φ → (ψyψ))
5 cbvexd.3 . 2 (φ → (x = y → (ψχ)))
62, 4, 5cbvexdh 1798 1 (φ → (xψyχ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1346  ∃wex 1378 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347 This theorem is referenced by:  cbvexdva  1801  vtoclgft  2598  bdsepnft  9321  strcollnft  9414
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