Theorem ral2imi 3068

Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3069. (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 3045	. 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 1815	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 3045	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		df-ral 3045	. . 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 1538   ∈ wcel 2109  ∀wral 3044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ral 3045

This theorem is referenced by:  ralim  3069  rexim  3070  ralbi  3085  r19.26  3091  iiner  8762  ss2ixp  8883  undifixp  8907  boxriin  8913  acni2  9999  axcc4  10392  intgru  10767  ingru  10768  prdsdsval3  17448  mndind  18755  hauscmplem  23293  uspgr2wlkeq  29574  wlkp1lem8  29608  prdstotbnd  37788  mnuunid  44266