Theorem ralimi2 3066

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 1812	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3050	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3050	. 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 1539   ∈ wcel 2113  ∀wral 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-ral 3050

This theorem is referenced by:  ralimia  3068  tfi  7793  resixpfo  8872  omex  9550  kmlem1  10059  brdom5  10437  brdom4  10438  xrub  13225  pcmptcl  16817  itgeq2  25733  iblcnlem  25744  pntrsumbnd  27531  nmounbseqi  30801  nmounbseqiALT  30802  sumdmdi  32444  dmdbr4ati  32445  dmdbr6ati  32447  bnj110  34963  fiinfi  43756