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Metamath Proof Explorer

Theorem raleqd

Description: Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses raleqd.a ⊢ Ⅎ 𝑥 𝐴
raleqd.b ⊢ Ⅎ 𝑥 𝐵
raleqd.e ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion raleqd ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step Hyp Ref Expression
1 raleqd.a ⊢ Ⅎ 𝑥 𝐴
2 raleqd.b ⊢ Ⅎ 𝑥 𝐵
3 raleqd.e ⊢ ( 𝜑 → 𝐴 = 𝐵 )
4 1 2 raleqf ⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
5 3 4 syl ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )