Metamath Proof Explorer


Theorem 3ralbidv

Description: Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 20-Feb-2025)

Ref Expression
Hypothesis 3ralbidv.1 ( 𝜑 → ( 𝜓𝜒 ) )
Assertion 3ralbidv ( 𝜑 → ( ∀ 𝑥𝐴𝑦𝐵𝑧𝐶 𝜓 ↔ ∀ 𝑥𝐴𝑦𝐵𝑧𝐶 𝜒 ) )

Proof

Step Hyp Ref Expression
1 3ralbidv.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 1 ralbidv ( 𝜑 → ( ∀ 𝑧𝐶 𝜓 ↔ ∀ 𝑧𝐶 𝜒 ) )
3 2 2ralbidv ( 𝜑 → ( ∀ 𝑥𝐴𝑦𝐵𝑧𝐶 𝜓 ↔ ∀ 𝑥𝐴𝑦𝐵𝑧𝐶 𝜒 ) )