Theorem r19.26m 1291

Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers.

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		19.26 749	. 2 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → ψ)))
2		df-ral 1205	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
3		df-ral 1205	. . 3 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
4	2, 3	anbi12i 369	. 2 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ B ψ) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → ψ)))
5	1, 4	bitr4 154	1 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x ∈ A φ ∧ ∀x ∈ B ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  tfrlem5 2953

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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