Theorem ralimdv2 3103

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)

Hypothesis

Ref	Expression
ralimdv2.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐵 → 𝜒)))

Assertion

Ref	Expression
ralimdv2	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem ralimdv2

Step	Hyp	Ref	Expression
1		ralimdv2.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐵 → 𝜒)))
2	1	alimdv 1922	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
3		df-ral 3070	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 3070	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
5	2, 3, 4	3imtr4g 295	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2109  ∀wral 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916

This theorem depends on definitions:  df-bi 206  df-ral 3070

This theorem is referenced by:  ralimdva  3104  ssralv  3991  zorn2lem7  10242  pwfseqlem3  10400  sup2  11914  xrsupexmnf  13021  xrinfmexpnf  13022  xrsupsslem  13023  xrinfmsslem  13024  xrub  13028  r19.29uz  15043  rexuzre  15045  caurcvg  15369  caucvg  15371  isprm5  16393  prmgaplem5  16737  prmgaplem6  16738  mrissmrid  17331  elcls3  22215  iscnp4  22395  cncls2  22405  cnntr  22407  2ndcsep  22591  dyadmbllem  24744  xrlimcnp  26099  pntlem3  26738  sigaclfu2  32068  lfuhgr2  33059  rdgssun  35528  mapdordlem2  39630  sn-sup2  40419  dffltz  40451  iunrelexp0  41263  climrec  43098  0ellimcdiv  43144