Theorem r19.27av 1293

Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

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

Distinct variable group(s):   ψ,x

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		pm2.27 30	. . . . 5 ⊢ (x ∈ A → ((x ∈ A → φ) → φ))
2	1	anim1d 432	. . . 4 ⊢ (x ∈ A → (((x ∈ A → φ) ∧ ψ) → (φ ∧ ψ)))
3	2	com12 13	. . 3 ⊢ (((x ∈ A → φ) ∧ ψ) → (x ∈ A → (φ ∧ ψ)))
4	3	19.20i 691	. 2 ⊢ (∀x((x ∈ A → φ) ∧ ψ) → ∀x(x ∈ A → (φ ∧ ψ)))
5		df-ral 1205	. . . 4 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6	5	anbi1i 368	. . 3 ⊢ ((∀x ∈ A φ ∧ ψ) ↔ (∀x(x ∈ A → φ) ∧ ψ))
7		19.27v 956	. . 3 ⊢ (∀x((x ∈ A → φ) ∧ ψ) ↔ (∀x(x ∈ A → φ) ∧ ψ))
8	6, 7	bitr4 154	. 2 ⊢ ((∀x ∈ A φ ∧ ψ) ↔ ∀x((x ∈ A → φ) ∧ ψ))
9		df-ral 1205	. 2 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ ∀x(x ∈ A → (φ ∧ ψ)))
10	4, 8, 9	3imtr4 192	1 ⊢ ((∀x ∈ A φ ∧ ψ) → ∀x ∈ A (φ ∧ ψ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  r19.28av 1294  spanun 5450

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

metamath.org