Theorem r19.28zv 1769

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty.

Assertion

Ref	Expression
r19.28zv	⊢ (¬ A = ∅ → (∀x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀x ∈ A ψ)))

Distinct variable group(s):   x,A   φ,x

Proof of Theorem r19.28zv

Step	Hyp	Ref	Expression
1		r19.3rzv 1767	. . 3 ⊢ (¬ A = ∅ → (φ ↔ ∀x ∈ A φ))
2	1	anbi1d 469	. 2 ⊢ (¬ A = ∅ → ((φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ)))
3		r19.26 1289	. 2 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ))
4	2, 3	syl6rbbr 417	1 ⊢ (¬ A = ∅ → (∀x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀x ∈ A ψ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ∀wral 1201  ∅c0 1707

This theorem is referenced by:  iindif2 2033

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-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

metamath.org