Theorem reximdvva 3019

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

Hypothesis

Ref	Expression
reximdvva.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))

Assertion

Ref	Expression
reximdvva	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝜑

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem reximdvva

Step	Hyp	Ref	Expression
1		reximdvva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))
2	1	anassrs 680	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	reximdva 3017	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒))
4	3	reximdva 3017	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  lcmgcdlem  15319  lsmelval2  19085  cpmadugsum  20683  axpasch  25821  frgrwopreglem5  27185  frgrwopreglem5ALT  27186  eulerpartlemgvv  30438  cvmlift2lem10  31294  ftc1anclem6  33490