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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-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:  cbvrexdva2  2840
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