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Theorem ralanid 3065
Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018.)
Assertion
Ref Expression
ralanid (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem ralanid
StepHypRef Expression
1 anclb 541 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 → (𝑥𝐴𝜑)))
21albii 1914 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
3 df-ral 3060 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 3060 . 2 (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
52, 3, 43bitr4ri 295 1 (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wcel 2155  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385  df-ral 3060
This theorem is referenced by:  idinxpssinxp2  34518
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