Theorem wl-equsb3 34786

Description: equsb3 2105 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 2152	. . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
2		nfeqf2 2391	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
3		equequ1 2028	. . . . 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 2541	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧))
6	5	sbbidv 2080	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
7		sbco2vv 2104	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
8		equsb3 2105	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧)
9	6, 7, 8	3bitr3g 315	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1531  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066

This theorem is referenced by: (None)