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Theorem bdbi 11186
Description: A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypotheses
Ref Expression
bdbi.1 BOUNDED 𝜑
bdbi.2 BOUNDED 𝜓
Assertion
Ref Expression
bdbi BOUNDED (𝜑𝜓)

Proof of Theorem bdbi
StepHypRef Expression
1 bdbi.1 . . . 4 BOUNDED 𝜑
2 bdbi.2 . . . 4 BOUNDED 𝜓
31, 2ax-bdim 11174 . . 3 BOUNDED (𝜑𝜓)
42, 1ax-bdim 11174 . . 3 BOUNDED (𝜓𝜑)
53, 4ax-bdan 11175 . 2 BOUNDED ((𝜑𝜓) ∧ (𝜓𝜑))
6 dfbi2 380 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
75, 6bd0r 11185 1 BOUNDED (𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  BOUNDED wbd 11172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-bd0 11173  ax-bdim 11174  ax-bdan 11175
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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