Theorem elimdelov

Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). (Contributed by Paul Chapman, 25-Mar-2008)

	Ref	Expression
Hypotheses	elimdelov.1	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐹 𝐵 ) )
	elimdelov.2	⊢ 𝑍 ∈ ( 𝑋 𝐹 𝑌 )
Assertion	elimdelov	⊢ if ( 𝜑 , 𝐶 , 𝑍 ) ∈ ( if ( 𝜑 , 𝐴 , 𝑋 ) 𝐹 if ( 𝜑 , 𝐵 , 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		elimdelov.1	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐹 𝐵 ) )
2		elimdelov.2	⊢ 𝑍 ∈ ( 𝑋 𝐹 𝑌 )
3		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐶 , 𝑍 ) = 𝐶 )
4		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝑋 ) = 𝐴 )
5		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐵 , 𝑌 ) = 𝐵 )
6	4 5	oveq12d	⊢ ( 𝜑 → ( if ( 𝜑 , 𝐴 , 𝑋 ) 𝐹 if ( 𝜑 , 𝐵 , 𝑌 ) ) = ( 𝐴 𝐹 𝐵 ) )
7	1 3 6	3eltr4d	⊢ ( 𝜑 → if ( 𝜑 , 𝐶 , 𝑍 ) ∈ ( if ( 𝜑 , 𝐴 , 𝑋 ) 𝐹 if ( 𝜑 , 𝐵 , 𝑌 ) ) )
8		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐶 , 𝑍 ) = 𝑍 )
9	8 2	eqeltrdi	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐶 , 𝑍 ) ∈ ( 𝑋 𝐹 𝑌 ) )
10		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝑋 ) = 𝑋 )
11		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐵 , 𝑌 ) = 𝑌 )
12	10 11	oveq12d	⊢ ( ¬ 𝜑 → ( if ( 𝜑 , 𝐴 , 𝑋 ) 𝐹 if ( 𝜑 , 𝐵 , 𝑌 ) ) = ( 𝑋 𝐹 𝑌 ) )
13	9 12	eleqtrrd	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐶 , 𝑍 ) ∈ ( if ( 𝜑 , 𝐴 , 𝑋 ) 𝐹 if ( 𝜑 , 𝐵 , 𝑌 ) ) )
14	7 13	pm2.61i	⊢ if ( 𝜑 , 𝐶 , 𝑍 ) ∈ ( if ( 𝜑 , 𝐴 , 𝑋 ) 𝐹 if ( 𝜑 , 𝐵 , 𝑌 ) )