Theorem wl-ax11-lem10

Description: We now have prepared everything. The unwanted variable u is just in one place left. pm2.61 can be used in conjunction with wl-ax11-lem9 to eliminate the second antecedent. Missing is something along the lines of ax-6 , so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019)

	Ref	Expression
Assertion	wl-ax11-lem10	⊢ ( ∀ 𝑦 𝑦 = 𝑢 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-ax11-lem8	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 𝜑 ) )
2		wl-ax11-lem6	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )
3	1 2	bitr3d	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑦 ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )
4	3	biimpd	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )
5	4	ex	⊢ ( ∀ 𝑢 𝑢 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) ) )
6	5	aecoms	⊢ ( ∀ 𝑦 𝑦 = 𝑢 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) ) )