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Theorem wl-ax11-lem9 33500
Description: The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem9 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))

Proof of Theorem wl-ax11-lem9
StepHypRef Expression
1 biidd 252 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜑))
21dral1 2356 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
32aecoms 2345 . . 3 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
43dral1 2356 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))
54aecoms 2345 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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