This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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