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Theorem wl-sbcom2d-lem2 33575
Description: Lemma used to prove wl-sbcom2d 33576. (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 ↔ ∀𝑥𝑦((𝑥 = 𝑢𝑦 = 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
StepHypRef Expression
1 id 22 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑥)
2 wl-naev 33534 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑣)
3 wl-naev 33534 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑢)
4 wl-naev 33534 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑢)
51, 2, 3, 4wl-2sb6d 33573 1 (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) → 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1594  [wsb 2010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011
This theorem is referenced by:  wl-sbcom2d  33576
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