Theorem ral2imi 2976

Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 2977. (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 2946	. 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 1783	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 2946	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		df-ral 2946	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
7	4, 5, 6	3imtr4g 285	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
8	1, 7	sylbi 207	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wal 1521   ∈ wcel 2030  ∀wral 2941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777

This theorem depends on definitions:  df-bi 197  df-ral 2946

This theorem is referenced by:  ralim  2977  rexim  3037  r19.26  3093  iiner  7862  ss2ixp  7963  undifixp  7986  boxriin  7992  acni2  8907  axcc4  9299  intgru  9674  ingru  9675  prdsdsval3  16192  mrcmndind  17413  hauscmplem  21257  uspgr2wlkeq  26598  wlkp1lem8  26633  prdstotbnd  33723