Theorem wl-equsb3 38489

Description: equsb3 2139 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)

Assertion

Ref	Expression
wl-equsb3	⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧))

Proof of Theorem wl-equsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfna1 2188	. . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
2		nfeqf2 2406	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
3		equequ1 2057	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 = 𝑧 ↔ 𝑤 = 𝑧))
4	3	a1i 11	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧 ↔ 𝑤 = 𝑧)))
5	1, 2, 4	sbied 2532	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧))
6	5	sbbidv 2115	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
7		sbco2vv 2135	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
8		equsb3 2139	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧)
9	6, 7, 8	3bitr3g 316	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀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: (None)