Theorem ralimi2 3069

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 1811	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
5	2, 3, 4	3imtr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2109  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ral 3053

This theorem is referenced by:  ralimia  3071  ralcom3OLD  3088  tfi  7853  resixpfo  8955  omex  9662  kmlem1  10170  brdom5  10548  brdom4  10549  xrub  13333  pcmptcl  16916  itgeq2  25736  iblcnlem  25747  pntrsumbnd  27534  nmounbseqi  30763  nmounbseqiALT  30764  sumdmdi  32406  dmdbr4ati  32407  dmdbr6ati  32409  bnj110  34894  fiinfi  43564