Theorem r19.26m 2750

Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)

Assertion

Ref	Expression
r19.26m	⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x ∈ A φ ∧ ∀x ∈ B ψ))

Proof of Theorem r19.26m

Step	Hyp	Ref	Expression
1		19.26 1593	. 2 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → ψ)))
2		df-ral 2620	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
3		df-ral 2620	. . 3 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
4	2, 3	anbi12i 678	. 2 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ B ψ) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → ψ)))
5	1, 4	bitr4i 243	1 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x ∈ A φ ∧ ∀x ∈ B ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀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-an 360  df-ral 2620

This theorem is referenced by: (None)