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Theorem rexbida 3260
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
rexbida.1 𝑥𝜑
rexbida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexbida (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Proof of Theorem rexbida
StepHypRef Expression
1 rexbida.1 . . 3 𝑥𝜑
2 rexbida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 577 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3exbid 2212 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
5 df-rex 3061 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
6 df-rex 3061 . 2 (∃𝑥𝐴 𝜒 ↔ ∃𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 313 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wex 1774  wnf 1778  wcel 2099  wrex 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-nf 1779  df-rex 3061
This theorem is referenced by:  rexbid  3262  rexbidvaALT  3264  dfiun2gOLD  5039  iuneq12daf  32477  bnj1366  34674  glbconxN  39077  supminfrnmpt  45060  limsupre2mpt  45351  limsupre3mpt  45355  limsupreuzmpt  45360
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