Theorem r19.21t 2700

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)

Assertion

Ref	Expression
r19.21t	⊢ (Ⅎxφ → (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ)))

Proof of Theorem r19.21t

Step	Hyp	Ref	Expression
1		bi2.04 350	. . . 4 ⊢ ((x ∈ A → (φ → ψ)) ↔ (φ → (x ∈ A → ψ)))
2	1	albii 1566	. . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ ∀x(φ → (x ∈ A → ψ)))
3		19.21t 1795	. . 3 ⊢ (Ⅎxφ → (∀x(φ → (x ∈ A → ψ)) ↔ (φ → ∀x(x ∈ A → ψ))))
4	2, 3	syl5bb 248	. 2 ⊢ (Ⅎxφ → (∀x(x ∈ A → (φ → ψ)) ↔ (φ → ∀x(x ∈ A → ψ))))
5		df-ral 2620	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
6		df-ral 2620	. . 3 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
7	6	imbi2i 303	. 2 ⊢ ((φ → ∀x ∈ A ψ) ↔ (φ → ∀x(x ∈ A → ψ)))
8	4, 5, 7	3bitr4g 279	1 ⊢ (Ⅎxφ → (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ)))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   ∈ 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  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  r19.21  2701