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Theorem rexbida 3263
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 578 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3exbid 2208 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
5 df-rex 3065 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
6 df-rex 3065 . 2 (∃𝑥𝐴 𝜒 ↔ ∃𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 314 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1773  wnf 1777  wcel 2098  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-rex 3065
This theorem is referenced by:  rexbid  3265  rexbidvaALT  3267  dfiun2gOLD  5027  iuneq12daf  32293  bnj1366  34369  glbconxN  38760  supminfrnmpt  44708  limsupre2mpt  44999  limsupre3mpt  45003  limsupreuzmpt  45008
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