Theorem r19.28av 2754

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.28av	⊢ ((φ ∧ ∀x ∈ A ψ) → ∀x ∈ A (φ ∧ ψ))

Distinct variable group:   φ,x

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

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 2753	. 2 ⊢ ((∀x ∈ A ψ ∧ φ) → ∀x ∈ A (ψ ∧ φ))
2		ancom 437	. 2 ⊢ ((φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A ψ ∧ φ))
3		ancom 437	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
4	3	ralbii 2639	. 2 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ ∀x ∈ A (ψ ∧ φ))
5	1, 2, 4	3imtr4i 257	1 ⊢ ((φ ∧ ∀x ∈ A ψ) → ∀x ∈ A (φ ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 358  ∀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-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  rr19.28v  2982  fununi  5161