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Metamath Proof Explorer

Theorem raleqtrdv

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025)

Ref Expression
Hypotheses raleqtrdv.1 ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
raleqtrdv.2 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion raleqtrdv ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step Hyp Ref Expression
1 raleqtrdv.1 ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
2 raleqtrdv.2 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
3 2 raleqdv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
4 1 3 mpbid ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )