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Theorem bdth 11191
Description: A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
Hypothesis
Ref Expression
bdth.1 𝜑
Assertion
Ref Expression
bdth BOUNDED 𝜑

Proof of Theorem bdth
StepHypRef Expression
1 ax-bdeq 11180 . . 3 BOUNDED 𝑥 = 𝑥
21, 1ax-bdim 11174 . 2 BOUNDED (𝑥 = 𝑥𝑥 = 𝑥)
3 id 19 . . 3 (𝑥 = 𝑥𝑥 = 𝑥)
4 bdth.1 . . 3 𝜑
53, 42th 172 . 2 ((𝑥 = 𝑥𝑥 = 𝑥) ↔ 𝜑)
62, 5bd0 11184 1 BOUNDED 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  BOUNDED wbd 11172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106  ax-bd0 11173  ax-bdim 11174  ax-bdeq 11180
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bdtru  11192  bdcvv  11217
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