Theorem difrab

Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004)

	Ref	Expression
Assertion	difrab	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∖ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) = { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∧ ¬ 𝜓 ) }

Proof

Step	Hyp	Ref	Expression
1		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
2		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) }
3	1 2	difeq12i	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∖ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) = ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∖ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } )
4		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∧ ¬ 𝜓 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ ¬ 𝜓 ) ) }
5		difab	⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∖ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } ) = { 𝑥 ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) }
6		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ ¬ 𝜓 ) ) )
7		simpr	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝜓 )
8	7	con3i	⊢ ( ¬ 𝜓 → ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
9	8	anim2i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ 𝜓 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
10		pm3.2	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜓 → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
11	10	adantr	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝜓 → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
12	11	con3d	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → ¬ 𝜓 ) )
13	12	imdistani	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ 𝜓 ) )
14	9 13	impbii	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ 𝜓 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
15	6 14	bitr3i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ ¬ 𝜓 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
16	15	abbii	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ ¬ 𝜓 ) ) } = { 𝑥 ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) }
17	5 16	eqtr4i	⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∖ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ ¬ 𝜓 ) ) }
18	4 17	eqtr4i	⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∧ ¬ 𝜓 ) } = ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∖ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } )
19	3 18	eqtr4i	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∖ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) = { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∧ ¬ 𝜓 ) }