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Metamath Proof Explorer

Theorem rabeqdv

Description: Equality of restricted class abstractions. Deduction form of rabeq . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypothesis rabeqdv.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion rabeqdv ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )

Proof

Step Hyp Ref Expression
1 rabeqdv.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 rabeq ⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )
3 1 2 syl ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )