Theorem wl-sbcom2d-lem1 38492

Description: Lemma used to prove wl-sbcom2d 38494. (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 2188	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑤
2		nfeqf2 2406	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑤 → Ⅎ𝑥 𝑣 = 𝑤)
3	1, 2	nfan1 2236	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤)
4		sbequ 2119	. . . . . 6 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
5	4	adantl 487	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
6	3, 5	sbbid 2281	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
7	6	ancoms 464	. . 3 ⊢ ((𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
8		sbequ 2119	. . 3 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
9	7, 8	sylan9bbr 520	. 2 ⊢ ((𝑢 = 𝑦 ∧ (𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤)) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
10	9	expr 462	1 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099

This theorem is used by:  wl-sbcom2d  38494