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Theorem bdcdeq 10788
Description: Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdcdeq.1 BOUNDED 𝜑
Assertion
Ref Expression
bdcdeq BOUNDED CondEq(𝑥 = 𝑦𝜑)

Proof of Theorem bdcdeq
StepHypRef Expression
1 ax-bdeq 10769 . . 3 BOUNDED 𝑥 = 𝑦
2 bdcdeq.1 . . 3 BOUNDED 𝜑
31, 2ax-bdim 10763 . 2 BOUNDED (𝑥 = 𝑦𝜑)
4 df-cdeq 2800 . 2 (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
53, 4bd0r 10774 1 BOUNDED CondEq(𝑥 = 𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  CondEqwcdeq 2799  BOUNDED wbd 10761
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 10762  ax-bdim 10763  ax-bdeq 10769
This theorem depends on definitions:  df-bi 115  df-cdeq 2800
This theorem is referenced by: (None)
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