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Theorem cbvexdva 1782
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 1398 . 2 yφ
2 nfvd 1399 . 2 (φ → Ⅎyψ)
3 cbvaldva.1 . . 3 ((φ x = y) → (ψχ))
43ex 108 . 2 (φ → (x = y → (ψχ)))
51, 2, 4cbvexd 1780 1 (φ → (xψyχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1358
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-nf 1326
This theorem is referenced by:  cbvrexdva2  2512  acexmid  5431  tfrlemi1  5863  ltexpri  6444  recexpr  6466
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