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Theorem wl-ax11-lem5 37590
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 2252 . . 3 (𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
21sps 2185 . 2 (∀𝑢 𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
32dral1 2444 1 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065
This theorem is referenced by:  wl-ax11-lem6  37591
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