Theorem bdcdeq 16865

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 16846	. . 3 |- BOUNDED x = y
2		bdcdeq.1	. . 3 |- BOUNDED ph
3	1, 2	ax-bdim 16840	. 2 |- BOUNDED ( x = y -> ph )
4		df-cdeq 3035	. 2 |- (CondEq ( x = y -> ph ) <-> ( x = y -> ph ) )
5	3, 4	bd0r 16851	1 |- BOUNDED CondEq ( x = y -> ph )

This proof depends on syntax axioms:   -> wi 4  CondEqwcdeq 3034  BOUNDED wbd 16838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 16839  ax-bdim 16840  ax-bdeq 16846

This proof depends on definitions:  df-bi 117  df-cdeq 3035

This theorem is used by: (None)