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Theorem reximdva0m 3230
 Description: Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
Hypothesis
Ref Expression
reximdva0m.1 ((φ x A) → ψ)
Assertion
Ref Expression
reximdva0m ((φ x x A) → x A ψ)
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem reximdva0m
StepHypRef Expression
1 reximdva0m.1 . . . . . 6 ((φ x A) → ψ)
21ex 108 . . . . 5 (φ → (x Aψ))
32ancld 308 . . . 4 (φ → (x A → (x A ψ)))
43eximdv 1757 . . 3 (φ → (x x Ax(x A ψ)))
54imp 115 . 2 ((φ x x A) → x(x A ψ))
6 df-rex 2306 . 2 (x A ψx(x A ψ))
75, 6sylibr 137 1 ((φ x x A) → x A ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301 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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-rex 2306 This theorem is referenced by: (None)
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