Theorem wl-sbcom2d-lem1 36728

Description: Lemma used to prove wl-sbcom2d 36730. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
wl-sbcom2d-lem1	⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 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

Step	Hyp	Ref	Expression
1		nfna1 2148	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑤
2		nfeqf2 2375	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑤 → Ⅎ𝑥 𝑣 = 𝑤)
3	1, 2	nfan1 2192	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤)
4		sbequ 2085	. . . . . 6 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
5	4	adantl 481	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
6	3, 5	sbbid 2237	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
7	6	ancoms 458	. . 3 ⊢ ((𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
8		sbequ 2085	. . 3 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
9	7, 8	sylan9bbr 510	. 2 ⊢ ((𝑢 = 𝑦 ∧ (𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤)) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
10	9	expr 456	1 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538  [wsb 2066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1781  df-nf 1785  df-sb 2067

This theorem is referenced by:  wl-sbcom2d  36730