Theorem r19.27z 3649

Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)

Hypothesis

Ref	Expression
r19.27z.1	⊢ Ⅎxψ

Assertion

Ref	Expression
r19.27z	⊢ (A ≠ ∅ → (∀x ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ψ)))

Distinct variable group:   x,A

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

Proof of Theorem r19.27z

Step	Hyp	Ref	Expression
1		r19.27z.1	. . . 4 ⊢ Ⅎxψ
2	1	r19.3rz 3642	. . 3 ⊢ (A ≠ ∅ → (ψ ↔ ∀x ∈ A ψ))
3	2	anbi2d 684	. 2 ⊢ (A ≠ ∅ → ((∀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	syl6rbbr 255	1 ⊢ (A ≠ ∅ → (∀x ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ≠ wne 2517  ∀wral 2615  ∅c0 3551

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  raaan  3658