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Theorem bd0r 11146
Description: A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 11145) biconditional in the hypothesis, to work better with definitions (𝜓 is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
Hypotheses
Ref Expression
bd0r.min BOUNDED 𝜑
bd0r.maj (𝜓𝜑)
Assertion
Ref Expression
bd0r BOUNDED 𝜓

Proof of Theorem bd0r
StepHypRef Expression
1 bd0r.min . 2 BOUNDED 𝜑
2 bd0r.maj . . 3 (𝜓𝜑)
32bicomi 130 . 2 (𝜑𝜓)
41, 3bd0 11145 1 BOUNDED 𝜓
Colors of variables: wff set class
Syntax hints:  wb 103  BOUNDED wbd 11133
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 11134
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bdbi  11147  bdstab  11148  bddc  11149  bd3or  11150  bd3an  11151  bdfal  11154  bdxor  11157  bj-bdcel  11158  bdab  11159  bdcdeq  11160  bdne  11174  bdnel  11175  bdreu  11176  bdrmo  11177  bdsbcALT  11180  bdss  11185  bdeq0  11188  bdvsn  11195  bdop  11196  bdeqsuc  11202  bj-bdind  11255
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