Theorem ralim 2686

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)

Assertion

Ref	Expression
ralim	⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A φ → ∀x ∈ A ψ))

Proof of Theorem ralim

Step	Hyp	Ref	Expression
1		df-ral 2620	. . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
2		ax-2 7	. . . 4 ⊢ ((x ∈ A → (φ → ψ)) → ((x ∈ A → φ) → (x ∈ A → ψ)))
3	2	al2imi 1561	. . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) → (∀x(x ∈ A → φ) → ∀x(x ∈ A → ψ)))
4	1, 3	sylbi 187	. 2 ⊢ (∀x ∈ A (φ → ψ) → (∀x(x ∈ A → φ) → ∀x(x ∈ A → ψ)))
5		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		df-ral 2620	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
7	4, 5, 6	3imtr4g 261	1 ⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A φ → ∀x ∈ A ψ))

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  ∀wral 2615

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 2620

This theorem is referenced by:  ral2imi  2691  r19.30  2757