Theorem ax8dfeq

Description: A version of ax-8 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010)

	Ref	Expression
Assertion	ax8dfeq	⊢ ∃ 𝑧 ( ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) → ( 𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		ax6e	⊢ ∃ 𝑧 𝑧 = 𝑤
2		ax8	⊢ ( 𝑤 = 𝑧 → ( 𝑤 ∈ 𝑥 → 𝑧 ∈ 𝑥 ) )
3	2	equcoms	⊢ ( 𝑧 = 𝑤 → ( 𝑤 ∈ 𝑥 → 𝑧 ∈ 𝑥 ) )
4		ax8	⊢ ( 𝑧 = 𝑤 → ( 𝑧 ∈ 𝑦 → 𝑤 ∈ 𝑦 ) )
5	3 4	imim12d	⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) → ( 𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦 ) ) )
6	1 5	eximii	⊢ ∃ 𝑧 ( ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) → ( 𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦 ) )