Theorem cdeqi 3722

Description: Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
cdeqi.1	⊢ (𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
cdeqi	⊢ CondEq(𝑥 = 𝑦 → 𝜑)

Proof of Theorem cdeqi

Step	Hyp	Ref	Expression
1		cdeqi.1	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
2		df-cdeq 3721	. 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
3	1, 2	mpbir 234	1 ⊢ CondEq(𝑥 = 𝑦 → 𝜑)

This proof depends on syntax axioms:   → wi 4  CondEqwcdeq 3720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-cdeq 3721

This theorem is used by:  cdeqth  3724  cdeqnot  3725  cdeqal  3726  cdeqab  3727  cdeqim  3730  cdeqcv  3731  cdeqeq  3732  cdeqel  3733