Theorem wl-sbcom2d-lem2 32522

Description: Lemma used to prove wl-sbcom2d 32523. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
wl-sbcom2d-lem2	⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑)))

Distinct variable groups:   𝑣,𝑢,𝑥   𝑦,𝑢,𝑣   𝜑,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-sbcom2d-lem2

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑥)
2		wl-naev 32481	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑣)
3		wl-naev 32481	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑢)
4		wl-naev 32481	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑢)
5	1, 2, 3, 4	wl-2sb6d 32520	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  [wsb 1867

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-11 2021  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  df-sb 1868

This theorem is referenced by:  wl-sbcom2d  32523