Theorem rabbiiaOLD

Description: Obsolete version of rabbiia as of 12-Jan-2025. (Contributed by NM, 22-May-1999) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	rabbiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rabbiiaOLD	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		rabbiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	pm5.32i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
3	2	abbii	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) }
4		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
5		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) }
6	3 4 5	3eqtr4i	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 }