Theorem r19.28av 1294

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 1293	. 2 ⊢ ((∀x ∈ A ψ ∧ φ) → ∀x ∈ A (ψ ∧ φ))
2		ancom 333	. 2 ⊢ ((φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A ψ ∧ φ))
3		ancom 333	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
4	3	biral 1223	. 2 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ ∀x ∈ A (ψ ∧ φ))
5	1, 2, 4	3imtr4 192	1 ⊢ ((φ ∧ ∀x ∈ A ψ) → ∀x ∈ A (φ ∧ ψ))

Syntax hints:   → wi 2   ∧ 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  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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