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Theorem wl-ax11-lem9 32549
 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 251 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜑))
21dral1 2313 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
32aecoms 2300 . . 3 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
43dral1 2313 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))
54aecoms 2300 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473 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 This theorem is referenced by: (None)
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