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Theorem rexbida 3261
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 571 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3exbid 2155 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
5 df-rex 3094 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
6 df-rex 3094 . 2 (∃𝑥𝐴 𝜒 ↔ ∃𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 306 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wex 1742  wnf 1746  wcel 2050  wrex 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-rex 3094
This theorem is referenced by:  rexbidvaALT  3262  rexbid  3263  dfiun2g  4825  dfiun2gOLD  4826  fun11iun  7458  iuneq12daf  30077  bnj1366  31755  glbconxN  35965  supminfrnmpt  41156  limsupre2mpt  41448  limsupre3mpt  41452  limsupreuzmpt  41457
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