Theorem ral2imi 2690

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 2689	. 2 ⊢ (∀x ∈ A φ → ∀x ∈ A (ψ → χ))
3		ralim 2685	. 2 ⊢ (∀x ∈ A (ψ → χ) → (∀x ∈ A ψ → ∀x ∈ A χ))
4	2, 3	syl 15	1 ⊢ (∀x ∈ A φ → (∀x ∈ A ψ → ∀x ∈ A χ))

Syntax hints:   → wi 4  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ral 2619

This theorem is referenced by:  rexim  2718  r19.26  2746