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Theorem cbvexdva 2011
 Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvaldva.1 ((φ x = y) → (ψχ))
Assertion
Ref Expression
cbvexdva (φ → (xψyχ))
Distinct variable groups:   ψ,y   χ,x   φ,x   φ,y
Allowed substitution hints:   ψ(x)   χ(y)

Proof of Theorem cbvexdva
StepHypRef Expression
1 nfv 1619 . 2 yφ
2 nfvd 1620 . 2 (φ → Ⅎyψ)
3 cbvaldva.1 . . 3 ((φ x = y) → (ψχ))
43ex 423 . 2 (φ → (x = y → (ψχ)))
51, 2, 4cbvexd 2009 1 (φ → (xψyχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541 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:  cbvrexdva2  2840
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