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Metamath Proof Explorer

Theorem rabbieq

Description: Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019)

	Ref	Expression
Hypotheses	rabbieq.1	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
	rabbieq.2	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	rabbieq	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		rabbieq.1	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
2		rabbieq.2	⊢ ( 𝜑 ↔ 𝜓 )
3	2	rabbii	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 }
4	1 3	eqtri	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜓 }