Theorem reximdvva 3214

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
ralimdvva.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		ralimdvva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))
2	1	anassrs 473	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	reximdva 3179	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒))
4	3	reximdva 3179	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3091

This theorem is used by:  reuop  6322  lcmgcdlem  16825  lsmelval2  21411  cpmadugsum  23251  mulsuniflem  28592  isplng  29313  axpasch  29576  frgrwopreglem5  30979  frgrwopreglem5ALT  30980  eulerpartlemgvv  35119  cusgr3cyclex  36109  cvmlift2lem10  36285  ftc1anclem6  38884  hashnexinjle  43445  prprelprb  48872  reupr  48877  grtriprop  49312