Theorem ral2imi 2562

Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)

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		ral2imi.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	ralimi 2560	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
3		ralim 2556	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
4	2, 3	syl 14	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463

This theorem depends on definitions:  df-bi 117  df-ral 2480

This theorem is referenced by:  r19.26  2623  r19.30dc  2644  iinerm  6666  ss2ixp  6770  bj-findis  15625