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Theorem cbvexdva 1801
 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 1418 . 2 yφ
2 nfvd 1419 . 2 (φ → Ⅎyψ)
3 cbvaldva.1 . . 3 ((φ x = y) → (ψχ))
43ex 108 . 2 (φ → (x = y → (ψχ)))
51, 2, 4cbvexd 1799 1 (φ → (xψyχ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃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:  cbvrexdva2  2532  acexmid  5454  tfrlemi1  5887  ltexpri  6586  recexpr  6609
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