Theorem bj-axc14nf

Description: Proof of a version of axc14 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axc14nf	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 𝑥 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nfeel2	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → Ⅎ 𝑧 𝑥 ∈ 𝑡 )
2		elequ2	⊢ ( 𝑡 = 𝑦 → ( 𝑥 ∈ 𝑡 ↔ 𝑥 ∈ 𝑦 ) )
3	1 2	bj-dvelimdv1	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 𝑥 ∈ 𝑦 ) )