Theorem ifsb

Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013)

	Ref	Expression
Hypotheses	ifsb.1	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 → 𝐶 = 𝐷 )
	ifsb.2	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 → 𝐶 = 𝐸 )
Assertion	ifsb	⊢ 𝐶 = if ( 𝜑 , 𝐷 , 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		ifsb.1	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 → 𝐶 = 𝐷 )
2		ifsb.2	⊢ ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 → 𝐶 = 𝐸 )
3		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
4	3 1	syl	⊢ ( 𝜑 → 𝐶 = 𝐷 )
5		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐷 , 𝐸 ) = 𝐷 )
6	4 5	eqtr4d	⊢ ( 𝜑 → 𝐶 = if ( 𝜑 , 𝐷 , 𝐸 ) )
7		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
8	7 2	syl	⊢ ( ¬ 𝜑 → 𝐶 = 𝐸 )
9		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐷 , 𝐸 ) = 𝐸 )
10	8 9	eqtr4d	⊢ ( ¬ 𝜑 → 𝐶 = if ( 𝜑 , 𝐷 , 𝐸 ) )
11	6 10	pm2.61i	⊢ 𝐶 = if ( 𝜑 , 𝐷 , 𝐸 )