Theorem bdcdeq 15495

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 = 𝑦 → 𝜑)

Proof of Theorem bdcdeq

Step	Hyp	Ref	Expression
1		ax-bdeq 15476	. . 3 ⊢ BOUNDED 𝑥 = 𝑦
2		bdcdeq.1	. . 3 ⊢ BOUNDED 𝜑
3	1, 2	ax-bdim 15470	. 2 ⊢ BOUNDED (𝑥 = 𝑦 → 𝜑)
4		df-cdeq 2973	. 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
5	3, 4	bd0r 15481	1 ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑)

Syntax hints:   → wi 4  CondEqwcdeq 2972  BOUNDED wbd 15468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 15469  ax-bdim 15470  ax-bdeq 15476

This theorem depends on definitions:  df-bi 117  df-cdeq 2973

This theorem is referenced by: (None)