Metamath Proof Explorer


Theorem reximdvva

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by AV, 5-Jan-2022)

Ref Expression
Hypothesis reximdvva.1 ( ( 𝜑 ∧ ( 𝑥𝐴𝑦𝐵 ) ) → ( 𝜓𝜒 ) )
Assertion reximdvva ( 𝜑 → ( ∃ 𝑥𝐴𝑦𝐵 𝜓 → ∃ 𝑥𝐴𝑦𝐵 𝜒 ) )

Proof

Step Hyp Ref Expression
1 reximdvva.1 ( ( 𝜑 ∧ ( 𝑥𝐴𝑦𝐵 ) ) → ( 𝜓𝜒 ) )
2 1 anassrs ( ( ( 𝜑𝑥𝐴 ) ∧ 𝑦𝐵 ) → ( 𝜓𝜒 ) )
3 2 reximdva ( ( 𝜑𝑥𝐴 ) → ( ∃ 𝑦𝐵 𝜓 → ∃ 𝑦𝐵 𝜒 ) )
4 3 reximdva ( 𝜑 → ( ∃ 𝑥𝐴𝑦𝐵 𝜓 → ∃ 𝑥𝐴𝑦𝐵 𝜒 ) )