Theorem ralimdv2 3089

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 1924	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
3		df-ral 3056	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 3056	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
5	2, 3, 4	3imtr4g 299	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4  ∀wal 1541   ∈ wcel 2112  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918

This theorem depends on definitions:  df-bi 210  df-ral 3056

This theorem is referenced by:  ralimdva  3090  ssralv  3953  zorn2lem7  10081  pwfseqlem3  10239  sup2  11753  xrsupexmnf  12860  xrinfmexpnf  12861  xrsupsslem  12862  xrinfmsslem  12863  xrub  12867  r19.29uz  14879  rexuzre  14881  caurcvg  15205  caucvg  15207  isprm5  16227  prmgaplem5  16571  prmgaplem6  16572  mrissmrid  17098  elcls3  21934  iscnp4  22114  cncls2  22124  cnntr  22126  2ndcsep  22310  dyadmbllem  24450  xrlimcnp  25805  pntlem3  26444  sigaclfu2  31755  lfuhgr2  32747  rdgssun  35235  mapdordlem2  39337  sn-sup2  40088  dffltz  40115  iunrelexp0  40928  climrec  42762  0ellimcdiv  42808