Theorem ifeq1

Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	ifeq1	⊢ ( 𝐴 = 𝐵 → if ( 𝜑 , 𝐴 , 𝐶 ) = if ( 𝜑 , 𝐵 , 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rabeq	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 } )
2	1	uneq1d	⊢ ( 𝐴 = 𝐵 → ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐶 ∣ ¬ 𝜑 } ) = ( { 𝑥 ∈ 𝐵 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐶 ∣ ¬ 𝜑 } ) )
3		dfif6	⊢ if ( 𝜑 , 𝐴 , 𝐶 ) = ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐶 ∣ ¬ 𝜑 } )
4		dfif6	⊢ if ( 𝜑 , 𝐵 , 𝐶 ) = ( { 𝑥 ∈ 𝐵 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐶 ∣ ¬ 𝜑 } )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → if ( 𝜑 , 𝐴 , 𝐶 ) = if ( 𝜑 , 𝐵 , 𝐶 ) )