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Theorem ralanid 34354
Description: Cancellation law for restriction. (Contributed by Peter Mazsa, 30-Dec-2018.)
Assertion
Ref Expression
ralanid (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem ralanid
StepHypRef Expression
1 anclb 571 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 → (𝑥𝐴𝜑)))
21albii 1896 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
3 df-ral 3056 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 3056 . 2 (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
52, 3, 43bitr4ri 293 1 (∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2140  wral 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886
This theorem depends on definitions:  df-bi 197  df-an 385  df-ral 3056
This theorem is referenced by:  idinxpssinxp2  34432
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