Theorem ral2imi 3069

Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3070. (Revised by Wolf Lammen, 1-Dec-2019.)

Hypothesis

Ref	Expression
ral2imi.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ral2imi	⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Proof of Theorem ral2imi

Step	Hyp	Ref	Expression
1		df-ral 3046	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		ral2imi.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imim3i 64	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → 𝜒)))
4	3	al2imi 1816	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 3046	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		df-ral 3046	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
7	4, 5, 6	3imtr4g 296	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
8	1, 7	sylbi 217	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2110  ∀wral 3045

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 3046

This theorem is referenced by:  ralim  3070  rexim  3071  ralbi  3085  r19.26  3090  iiner  8708  ss2ixp  8829  undifixp  8853  boxriin  8859  acni2  9929  axcc4  10322  intgru  10697  ingru  10698  prdsdsval3  17381  mndind  18728  hauscmplem  23314  uspgr2wlkeq  29617  wlkp1lem8  29650  prdstotbnd  37813  mnuunid  44289