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Theorem wl-sbcom2d-lem1 33673
Description: Lemma used to prove wl-sbcom2d 33675. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
Assertion
Ref Expression
wl-sbcom2d-lem1 ((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
Distinct variable groups:   𝑣,𝑢,𝑥   𝑦,𝑢,𝑣   𝑤,𝑢,𝑣   𝑧,𝑢,𝑣   𝜑,𝑢,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wl-sbcom2d-lem1
StepHypRef Expression
1 nfna1 2178 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑤
2 nfeqf2 2442 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑤 → Ⅎ𝑥 𝑣 = 𝑤)
31, 2nfan1 2215 . . . . 5 𝑥(¬ ∀𝑥 𝑥 = 𝑤𝑣 = 𝑤)
4 sbequ 2513 . . . . . 6 (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
54adantl 473 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑤𝑣 = 𝑤) → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
63, 5sbbid 2540 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑤𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
76ancoms 468 . . 3 ((𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
8 sbequ 2513 . . 3 (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
97, 8sylan9bbr 739 . 2 ((𝑢 = 𝑦 ∧ (𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤)) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
109expr 644 1 ((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630  [wsb 2046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047
This theorem is referenced by:  wl-sbcom2d  33675
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