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Theorem 3exbidv 1825
Description: Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
Hypothesis
Ref Expression
3exbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
3exbidv  |-  ( ph  ->  ( E. x E. y E. z ps  <->  E. x E. y E. z ch ) )
Distinct variable groups:    ph, x    ph, y    ph, z
Allowed substitution hints:    ps( x, y, z)    ch( x, y, z)

Proof of Theorem 3exbidv
StepHypRef Expression
1 3exbidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21exbidv 1781 . 2  |-  ( ph  ->  ( E. z ps  <->  E. z ch ) )
322exbidv 1824 1  |-  ( ph  ->  ( E. x E. y E. z ps  <->  E. x E. y E. z ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   E.wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ceqsex6v  2704  euotd  4146  oprabid  5771  eloprabga  5826  eloprabi  6062
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