Theorem ax5el

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ax5el	⊢ ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		ax-c14	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) ) )
2		ax-c16	⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) )
3		ax-c16	⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 ) )
4	1 2 3	pm2.61ii	⊢ ( 𝑥 ∈ 𝑦 → ∀ 𝑧 𝑥 ∈ 𝑦 )