Theorem spsbce-2

Description: Theorem *11.36 in WhiteheadRussell p. 162. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	spsbce-2	⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		spsbe	⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 → ∃ 𝑥 [ 𝑤 / 𝑦 ] 𝜑 )
2		spsbe	⊢ ( [ 𝑤 / 𝑦 ] 𝜑 → ∃ 𝑦 𝜑 )
3	2	eximi	⊢ ( ∃ 𝑥 [ 𝑤 / 𝑦 ] 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )
4	1 3	syl	⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )