Theorem ralimdaa 2692

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)

Hypotheses

Ref	Expression
ralimdaa.1	⊢ Ⅎxφ
ralimdaa.2	⊢ ((φ ∧ x ∈ A) → (ψ → χ))

Assertion

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

Proof of Theorem ralimdaa

Step	Hyp	Ref	Expression
1		ralimdaa.1	. . 3 ⊢ Ⅎxφ
2		ralimdaa.2	. . . . 5 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
3	2	ex 423	. . . 4 ⊢ (φ → (x ∈ A → (ψ → χ)))
4	3	a2d 23	. . 3 ⊢ (φ → ((x ∈ A → ψ) → (x ∈ A → χ)))
5	1, 4	alimd 1764	. 2 ⊢ (φ → (∀x(x ∈ A → ψ) → ∀x(x ∈ A → χ)))
6		df-ral 2620	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
7		df-ral 2620	. 2 ⊢ (∀x ∈ A χ ↔ ∀x(x ∈ A → χ))
8	5, 6, 7	3imtr4g 261	1 ⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ))

Syntax hints:   → wi 4   ∧ wa 358  ∀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-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralimdva  2693