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Theorem ralanid 3166
Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)
Assertion
Ref Expression
ralanid (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem ralanid
StepHypRef Expression
1 ibar 531 . . 3 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
21bicomd 225 . 2 (𝑥𝐴 → ((𝑥𝐴𝜑) ↔ 𝜑))
32ralbiia 3162 1 (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wcel 2107  wral 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 209  df-an 399  df-ral 3141
This theorem is referenced by:  idinxpssinxp2  35562
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