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Theorem wl-ax11-lem5 34691
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 2247 . . 3 (𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
21sps 2176 . 2 (∀𝑢 𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
32dral1 2458 1 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063
This theorem is referenced by:  wl-ax11-lem6  34692
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