Theorem ralimi2 3080

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 1819	. 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 295	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4  ∀wal 1541   ∈ wcel 2110  ∀wral 3061

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

This theorem depends on definitions:  df-bi 210  df-ral 3066

This theorem is referenced by:  ralimia  3081  ralcom3  3276  tfi  7632  resixpfo  8617  omex  9258  kmlem1  9764  brdom5  10143  brdom4  10144  xrub  12902  pcmptcl  16444  itgeq2  24675  iblcnlem  24686  pntrsumbnd  26447  nmounbseqi  28858  nmounbseqiALT  28859  sumdmdi  30501  dmdbr4ati  30502  dmdbr6ati  30504  bnj110  32551  fiinfi  40856