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Theorem wl-ax11-lem5 32545
 Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem5 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Proof of Theorem wl-ax11-lem5
StepHypRef Expression
1 sbequ12r 2098 . . 3 (𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
21sps 2043 . 2 (∀𝑢 𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
32dral1 2313 1 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  wl-ax11-lem6  32546
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