Theorem r19.26-2 1290

Description: Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers.

Assertion

Ref	Expression
r19.26-2	⊢ (∀x ∈ A ∀y ∈ B (φ ∧ ψ) ↔ (∀x ∈ A ∀y ∈ B φ ∧ ∀x ∈ A ∀y ∈ B ψ))

Proof of Theorem r19.26-2

Step	Hyp	Ref	Expression
1		r19.26 1289	. . 3 ⊢ (∀y ∈ B (φ ∧ ψ) ↔ (∀y ∈ B φ ∧ ∀y ∈ B ψ))
2	1	biral 1223	. 2 ⊢ (∀x ∈ A ∀y ∈ B (φ ∧ ψ) ↔ ∀x ∈ A (∀y ∈ B φ ∧ ∀y ∈ B ψ))
3		r19.26 1289	. 2 ⊢ (∀x ∈ A (∀y ∈ B φ ∧ ∀y ∈ B ψ) ↔ (∀x ∈ A ∀y ∈ B φ ∧ ∀x ∈ A ∀y ∈ B ψ))
4	2, 3	bitr 151	1 ⊢ (∀x ∈ A ∀y ∈ B (φ ∧ ψ) ↔ (∀x ∈ A ∀y ∈ B φ ∧ ∀x ∈ A ∀y ∈ B ψ))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wral 1201

This theorem is referenced by:  fununi 2705

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-or 197  df-an 198  df-ral 1205

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