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Theorem ralbid2 2461
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Hypotheses
Ref Expression
ralbid2.nf 𝑥𝜑
ralbid2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
ralbid2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Proof of Theorem ralbid2
StepHypRef Expression
1 ralbid2.nf . . 3 𝑥𝜑
2 ralbid2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
31, 2albid 1595 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐵𝜒)))
4 df-ral 2440 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
5 df-ral 2440 . 2 (∀𝑥𝐵 𝜒 ↔ ∀𝑥(𝑥𝐵𝜒))
63, 4, 53bitr4g 222 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1333  wnf 1440  wcel 2128  wral 2435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440
This theorem is referenced by:  strcollnft  13630
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