Theorem r19.28av 1758

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:   φ,x

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 1757	. 2 ⊢ ((∀x ∈ A ψ ⋀ φ) → ∀x ∈ A (ψ ⋀ φ))
2		ancom 437	. 2 ⊢ ((φ ⋀ ∀x ∈ A ψ) ↔ (∀x ∈ A ψ ⋀ φ))
3		ancom 437	. . 3 ⊢ ((φ ⋀ ψ) ↔ (ψ ⋀ φ))
4	3	ralbii 1670	. 2 ⊢ (∀x ∈ A (φ ⋀ ψ) ↔ ∀x ∈ A (ψ ⋀ φ))
5	1, 2, 4	3imtr4 219	1 ⊢ ((φ ⋀ ∀x ∈ A ψ) → ∀x ∈ A (φ ⋀ ψ))

Syntax hints:   → wi 3   ⋀ wa 223  ∀wral 1648

This theorem is referenced by:  fununi 3569  fsummulc1 7033  minveclem27 8567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1652

Copyright terms: Public domain