Theorem bj-axc14

Description: Alternate proof of axc14 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axc14	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-axc14nf	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 𝑥 ∈ 𝑦 ) )
2		nf5r	⊢ ( Ⅎ 𝑧 𝑥 ∈ 𝑦 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) )
3	2	a1i	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( Ⅎ 𝑧 𝑥 ∈ 𝑦 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) ) )
4	1 3	syld	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) ) )