Theorem 19.9d2rf

Description: A deduction version of one direction of 19.9 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017)

	Ref	Expression
Hypotheses	19.9d2rf.0	⊢ Ⅎ 𝑦 𝜑
	19.9d2rf.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
	19.9d2rf.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	19.9d2rf.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 )
Assertion	19.9d2rf	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		19.9d2rf.0	⊢ Ⅎ 𝑦 𝜑
2		19.9d2rf.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		19.9d2rf.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
4		19.9d2rf.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 )
5		rexex	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 → ∃ 𝑥 ∃ 𝑦 ∈ 𝐵 𝜓 )
6		rexex	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜓 → ∃ 𝑦 𝜓 )
7	6	eximi	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐵 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜓 )
8	4 5 7	3syl	⊢ ( 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 )
9	1 2	nfexd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 𝜓 )
10	9	19.9d	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) )
11	8 10	mpd	⊢ ( 𝜑 → ∃ 𝑦 𝜓 )
12	3	19.9d	⊢ ( 𝜑 → ( ∃ 𝑦 𝜓 → 𝜓 ) )
13	11 12	mpd	⊢ ( 𝜑 → 𝜓 )