Theorem bdcdeq 9959

Description: Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdcdeq.1	|- BOUNDED ph

Assertion

Ref	Expression
bdcdeq	|- BOUNDED CondEq ( x = y -> ph )

Proof of Theorem bdcdeq

Step	Hyp	Ref	Expression
1		ax-bdeq 9940	. . 3 |- BOUNDED x = y
2		bdcdeq.1	. . 3 |- BOUNDED ph
3	1, 2	ax-bdim 9934	. 2 |- BOUNDED ( x = y -> ph )
4		df-cdeq 2748	. 2 |- (CondEq ( x = y -> ph ) <-> ( x = y -> ph ) )
5	3, 4	bd0r 9945	1 |- BOUNDED CondEq ( x = y -> ph )

Syntax hints:   -> wi 4  CondEqwcdeq 2747  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-bd0 9933  ax-bdim 9934  ax-bdeq 9940

This theorem depends on definitions:  df-bi 110  df-cdeq 2748

This theorem is referenced by: (None)