Theorem wl-equsb4

Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019)

	Ref	Expression
Assertion	wl-equsb4	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		nfeqf	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑦 = 𝑧 )
2	1	ex	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 𝑦 = 𝑧 ) )
3		sbft	⊢ ( Ⅎ 𝑥 𝑦 = 𝑧 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) )
4	2 3	syl6com	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) ) )
5		sbequ12r	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) )
6	5	equcoms	⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) )
7	6	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) )
8	4 7	pm2.61d2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑧 ↔ 𝑦 = 𝑧 ) )