Theorem bdcdeq 13874

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 13855	. . 3 |- BOUNDED x = y
2		bdcdeq.1	. . 3 |- BOUNDED ph
3	1, 2	ax-bdim 13849	. 2 |- BOUNDED ( x = y -> ph )
4		df-cdeq 2939	. 2 |- (CondEq ( x = y -> ph ) <-> ( x = y -> ph ) )
5	3, 4	bd0r 13860	1 |- BOUNDED CondEq ( x = y -> ph )

Syntax hints:   -> wi 4  CondEqwcdeq 2938  BOUNDED wbd 13847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13848  ax-bdim 13849  ax-bdeq 13855

This theorem depends on definitions:  df-bi 116  df-cdeq 2939

This theorem is referenced by: (None)