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Theorem rexbidv2 2346
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
rexbidv2.1  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch ) ) )
Assertion
Ref Expression
rexbidv2  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem rexbidv2
StepHypRef Expression
1 rexbidv2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch ) ) )
21exbidv 1722 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  B  /\  ch ) ) )
3 df-rex 2329 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
4 df-rex 2329 . 2  |-  ( E. x  e.  B  ch  <->  E. x ( x  e.  B  /\  ch )
)
52, 3, 43bitr4g 216 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-rex 2329
This theorem is referenced by:  rexss  3035  rexsupp  5319  isoini  5485  ltexpi  6493  rexuz  8619
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