Theorem ralimi2 3099

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 1844	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3082	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3082	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
5	2, 3, 4	3imtr4i 295	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2146  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ral 3082

This theorem is used by:  ralimia  3101  tfi  7855  resixpfo  8940  omex  9619  kmlem1  10150  brdom5  10528  brdom4  10529  xrub  13356  pcmptcl  16975  itgeq2  25990  iblcnlem  26001  pntrsumbnd  27783  nmounbseqi  31202  nmounbseqiALT  31203  sumdmdi  32845  dmdbr4ati  32846  dmdbr6ati  32848  bnj110  35313  fiinfi  44359