Theorem r19.20 1251

Description: Distribution of restricted quantification over implication.

Assertion

Ref	Expression
r19.20	⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A φ → ∀x ∈ A ψ))

Proof of Theorem r19.20

Step	Hyp	Ref	Expression
1		df-ral 1205	. . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
2		ax-2 4	. . . 4 ⊢ ((x ∈ A → (φ → ψ)) → ((x ∈ A → φ) → (x ∈ A → ψ)))
3	2	19.20ii 692	. . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) → (∀x(x ∈ A → φ) → ∀x(x ∈ A → ψ)))
4	1, 3	sylbi 174	. 2 ⊢ (∀x ∈ A (φ → ψ) → (∀x(x ∈ A → φ) → ∀x(x ∈ A → ψ)))
5		df-ral 1205	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		df-ral 1205	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
7	4, 5, 6	3imtr4g 426	1 ⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A φ → ∀x ∈ A ψ))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  tfrlem1 2949  tz7.49 2997  abianfp 3000  bnd 3548  kmlem11 3590  osumlem4 5533

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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