Theorem ralimi2 3098

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)

Hypothesis

Ref	Expression
ralimi2.1	⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓))

Assertion

Ref	Expression
ralimi2	⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Proof of Theorem ralimi2

Step	Hyp	Ref	Expression
1		ralimi2.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓))
2	1	alimi 1887	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3066	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3066	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
5	2, 3, 4	3imtr4i 281	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4  ∀wal 1629   ∈ wcel 2145  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885

This theorem depends on definitions:  df-bi 197  df-ral 3066

This theorem is referenced by:  ralimia  3099  ralcom3  3253  tfi  7204  resixpfo  8104  omex  8708  kmlem1  9178  brdom5  9557  brdom4  9558  xrub  12347  pcmptcl  15802  itgeq2  23764  iblcnlem  23775  pntrsumbnd  25476  nmounbseqi  27972  nmounbseqiALT  27973  sumdmdi  29619  dmdbr4ati  29620  dmdbr6ati  29622  bnj110  31266  fiinfi  38402