Theorem elim2if

Description: Elimination of two conditional operators contained in a wff ch . (Contributed by Thierry Arnoux, 25-Jan-2017)

	Ref	Expression
Hypotheses	elim2if.1	⊢ ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐴 → ( 𝜒 ↔ 𝜃 ) )
	elim2if.2	⊢ ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐵 → ( 𝜒 ↔ 𝜏 ) )
	elim2if.3	⊢ ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐶 → ( 𝜒 ↔ 𝜂 ) )
Assertion	elim2if	⊢ ( 𝜒 ↔ ( ( 𝜑 ∧ 𝜃 ) ∨ ( ¬ 𝜑 ∧ ( ( 𝜓 ∧ 𝜏 ) ∨ ( ¬ 𝜓 ∧ 𝜂 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elim2if.1	⊢ ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐴 → ( 𝜒 ↔ 𝜃 ) )
2		elim2if.2	⊢ ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐵 → ( 𝜒 ↔ 𝜏 ) )
3		elim2if.3	⊢ ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐶 → ( 𝜒 ↔ 𝜂 ) )
4		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐴 )
5	4 1	syl	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
6		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = if ( 𝜓 , 𝐵 , 𝐶 ) )
7	6	eqeq1d	⊢ ( ¬ 𝜑 → ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐵 ↔ if ( 𝜓 , 𝐵 , 𝐶 ) = 𝐵 ) )
8	7 2	syl6bir	⊢ ( ¬ 𝜑 → ( if ( 𝜓 , 𝐵 , 𝐶 ) = 𝐵 → ( 𝜒 ↔ 𝜏 ) ) )
9	6	eqeq1d	⊢ ( ¬ 𝜑 → ( if ( 𝜑 , 𝐴 , if ( 𝜓 , 𝐵 , 𝐶 ) ) = 𝐶 ↔ if ( 𝜓 , 𝐵 , 𝐶 ) = 𝐶 ) )
10	9 3	syl6bir	⊢ ( ¬ 𝜑 → ( if ( 𝜓 , 𝐵 , 𝐶 ) = 𝐶 → ( 𝜒 ↔ 𝜂 ) ) )
11	8 10	elimifd	⊢ ( ¬ 𝜑 → ( 𝜒 ↔ ( ( 𝜓 ∧ 𝜏 ) ∨ ( ¬ 𝜓 ∧ 𝜂 ) ) ) )
12	5 11	cases	⊢ ( 𝜒 ↔ ( ( 𝜑 ∧ 𝜃 ) ∨ ( ¬ 𝜑 ∧ ( ( 𝜓 ∧ 𝜏 ) ∨ ( ¬ 𝜓 ∧ 𝜂 ) ) ) ) )