Theorem ral2imi 3084

Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3085. (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 3061	. 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 1817	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 3061	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		df-ral 3061	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
7	4, 5, 6	3imtr4g 295	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
8	1, 7	sylbi 216	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2106  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 206  df-ral 3061

This theorem is referenced by:  ralim  3085  rexim  3086  ralbi  3102  r19.26  3110  iiner  8735  ss2ixp  8855  undifixp  8879  boxriin  8885  acni2  9991  axcc4  10384  intgru  10759  ingru  10760  prdsdsval3  17381  mndind  18652  hauscmplem  22794  uspgr2wlkeq  28657  wlkp1lem8  28691  prdstotbnd  36326  mnuunid  42679