Theorem naev 2061

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
naev	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Distinct variable group:   𝑣,𝑢

Proof of Theorem naev

Step	Hyp	Ref	Expression
1		aev 2058	. 2 ⊢ (∀𝑢 𝑢 = 𝑣 → ∀𝑥 𝑥 = 𝑦)
2	1	con3i 154	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

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 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  naev2  2062  wl-sbcom2d-lem2  37583  wl-sbal1  37586  wl-sbal2  37587  wl-ax11-lem3  37610