Theorem 19.9d2r

Description: A deduction version of one direction of 19.9 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017)

	Ref	Expression
Hypotheses	19.9d2r.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
	19.9d2r.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	19.9d2r.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 )
Assertion	19.9d2r	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		19.9d2r.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
2		19.9d2r.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3		19.9d2r.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 )
4		nfv	⊢ Ⅎ 𝑦 𝜑
5	4 1 2 3	19.9d2rf	⊢ ( 𝜑 → 𝜓 )