Theorem wl-naev 32481

Description: If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.)

Assertion

Ref	Expression
wl-naev	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Distinct variable group:   𝑣,𝑢

Proof of Theorem wl-naev

Step	Hyp	Ref	Expression
1		aev 1970	. 2 ⊢ (∀𝑢 𝑢 = 𝑣 → ∀𝑥 𝑥 = 𝑦)
2	1	con3i 149	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Syntax hints:  ¬ wn 3   → wi 4  ∀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

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  wl-sbcom2d-lem2  32522  wl-sbal1  32525  wl-sbal2  32526  wl-ax11-lem3  32543