Theorem r19.27av 2753

Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.27av	⊢ ((∀x ∈ A φ ∧ ψ) → ∀x ∈ A (φ ∧ ψ))

Distinct variable group:   ψ,x

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 ⊢ (ψ → (x ∈ A → ψ))
2	1	ralrimiv 2697	. . 3 ⊢ (ψ → ∀x ∈ A ψ)
3	2	anim2i 552	. 2 ⊢ ((∀x ∈ A φ ∧ ψ) → (∀x ∈ A φ ∧ ∀x ∈ A ψ))
4		r19.26 2747	. 2 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ))
5	3, 4	sylibr 203	1 ⊢ ((∀x ∈ A φ ∧ ψ) → ∀x ∈ A (φ ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 358   ∈ 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:  r19.28av  2754