Theorem r19.27av 1752

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:   ψ,x

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		pm2.27 62	. . . . 5 ⊢ (x ∈ A → ((x ∈ A → φ) → φ))
2	1	anim1d 559	. . . 4 ⊢ (x ∈ A → (((x ∈ A → φ) ⋀ ψ) → (φ ⋀ ψ)))
3	2	com12 11	. . 3 ⊢ (((x ∈ A → φ) ⋀ ψ) → (x ∈ A → (φ ⋀ ψ)))
4	3	19.20i 991	. 2 ⊢ (∀x((x ∈ A → φ) ⋀ ψ) → ∀x(x ∈ A → (φ ⋀ ψ)))
5		df-ral 1647	. . . 4 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6	5	anbi1i 481	. . 3 ⊢ ((∀x ∈ A φ ⋀ ψ) ↔ (∀x(x ∈ A → φ) ⋀ ψ))
7		19.27v 1297	. . 3 ⊢ (∀x((x ∈ A → φ) ⋀ ψ) ↔ (∀x(x ∈ A → φ) ⋀ ψ))
8	6, 7	bitr4 176	. 2 ⊢ ((∀x ∈ A φ ⋀ ψ) ↔ ∀x((x ∈ A → φ) ⋀ ψ))
9		df-ral 1647	. 2 ⊢ (∀x ∈ A (φ ⋀ ψ) ↔ ∀x(x ∈ A → (φ ⋀ ψ)))
10	4, 8, 9	3imtr4 219	1 ⊢ ((∀x ∈ A φ ⋀ ψ) → ∀x ∈ A (φ ⋀ ψ))

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 953   ∈ wcel 957  ∀wral 1643

This theorem is referenced by:  r19.28av 1753  spanun 9422

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-17 970  ax-4 972  ax-5o 974

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1647

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