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Theorem ralbidb 45725
Description: Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 45726 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)
Hypotheses
Ref Expression
ralbidb.1 (𝜑 → (𝑥𝐴 ↔ (𝑥𝐵𝜓)))
ralbidb.2 ((𝜑𝑥𝐴) → (𝜒𝜃))
Assertion
Ref Expression
ralbidb (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 (𝜓𝜃)))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ralbidb
StepHypRef Expression
1 ralbidb.1 . . . 4 (𝜑 → (𝑥𝐴 ↔ (𝑥𝐵𝜓)))
2 ralbidb.2 . . . 4 ((𝜑𝑥𝐴) → (𝜒𝜃))
31, 2logic1a 45718 . . 3 (𝜑 → ((𝑥𝐴𝜒) ↔ ((𝑥𝐵𝜓) → 𝜃)))
4 impexp 454 . . 3 (((𝑥𝐵𝜓) → 𝜃) ↔ (𝑥𝐵 → (𝜓𝜃)))
53, 4bitrdi 290 . 2 (𝜑 → ((𝑥𝐴𝜒) ↔ (𝑥𝐵 → (𝜓𝜃))))
65ralbidv2 3108 1 (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  wral 3054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917
This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3059
This theorem is referenced by: (None)
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