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Metamath Proof Explorer

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 )