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Theorem cbvexd 2009
 Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbvald.1 yφ
cbvald.2 (φ → Ⅎyψ)
cbvald.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 cbvald.1 . . . 4 yφ
2 cbvald.2 . . . . 5 (φ → Ⅎyψ)
32nfnd 1791 . . . 4 (φ → Ⅎy ¬ ψ)
4 cbvald.3 . . . . 5 (φ → (x = y → (ψχ)))
5 notbi 286 . . . . 5 ((ψχ) ↔ (¬ ψ ↔ ¬ χ))
64, 5syl6ib 217 . . . 4 (φ → (x = y → (¬ ψ ↔ ¬ χ)))
71, 3, 6cbvald 2008 . . 3 (φ → (x ¬ ψy ¬ χ))
87notbid 285 . 2 (φ → (¬ x ¬ ψ ↔ ¬ y ¬ χ))
9 df-ex 1542 . 2 (xψ ↔ ¬ x ¬ ψ)
10 df-ex 1542 . 2 (yχ ↔ ¬ y ¬ χ)
118, 9, 103bitr4g 279 1 (φ → (xψyχ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 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 This theorem is referenced by:  cbvexdva  2011  vtoclgft  2905  dfid3  4768
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