Theorem rabbi

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii . (Contributed by NM, 25-Nov-2013)

	Ref	Expression
Assertion	rabbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) ↔ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		abbib	⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
2		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) }
3		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜒 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) }
4	2 3	eqeq12i	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } ↔ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) } )
5		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜓 ↔ 𝜒 ) ) )
6		pm5.32	⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
7	6	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜓 ↔ 𝜒 ) ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
8	5 7	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
9	1 4 8	3bitr4ri	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) ↔ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )