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Theorem rexbida 3280
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 582 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3exbid 2224 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
5 df-rex 3115 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
6 df-rex 3115 . 2 (∃𝑥𝐴 𝜒 ↔ ∃𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 317 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wnf 1785  wcel 2112  wrex 3110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-rex 3115
This theorem is referenced by:  rexbidvaALT  3281  rexbid  3282  ralrexbidOLD  3285  dfiun2g  4920  dfiun2gOLD  4921  iuneq12daf  30324  bnj1366  32215  glbconxN  36673  supminfrnmpt  42075  limsupre2mpt  42365  limsupre3mpt  42369  limsupreuzmpt  42374
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