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Theorem rexbida 3315
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 579 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3exbid 2215 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
5 df-rex 3141 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
6 df-rex 3141 . 2 (∃𝑥𝐴 𝜒 ↔ ∃𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 315 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wex 1771  wnf 1775  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-rex 3141
This theorem is referenced by:  rexbidvaALT  3316  rexbid  3317  ralrexbidOLD  3320  dfiun2g  4946  dfiun2gOLD  4947  iuneq12daf  30236  bnj1366  32000  glbconxN  36394  supminfrnmpt  41595  limsupre2mpt  41887  limsupre3mpt  41891  limsupreuzmpt  41896
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