Theorem wl-equsb3

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

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

Proof

Step	Hyp	Ref	Expression
1		nfna1	⊢ Ⅎ 𝑦 ¬ ∀ 𝑦 𝑦 = 𝑧
2		nfeqf2	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑤 = 𝑧 )
3		equequ1	⊢ ( 𝑦 = 𝑤 → ( 𝑦 = 𝑧 ↔ 𝑤 = 𝑧 ) )
4	3	a1i	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( 𝑦 = 𝑤 → ( 𝑦 = 𝑧 ↔ 𝑤 = 𝑧 ) ) )
5	1 2 4	sbied	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑤 / 𝑦 ] 𝑦 = 𝑧 ↔ 𝑤 = 𝑧 ) )
6	5	sbbidv	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑥 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝑦 = 𝑧 ↔ [ 𝑥 / 𝑤 ] 𝑤 = 𝑧 ) )
7		sbco2vv	⊢ ( [ 𝑥 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝑦 = 𝑧 ↔ [ 𝑥 / 𝑦 ] 𝑦 = 𝑧 )
8		equsb3	⊢ ( [ 𝑥 / 𝑤 ] 𝑤 = 𝑧 ↔ 𝑥 = 𝑧 )
9	6 7 8	3bitr3g	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( [ 𝑥 / 𝑦 ] 𝑦 = 𝑧 ↔ 𝑥 = 𝑧 ) )