Theorem r19.27v 3070

Description: Restricted quantitifer version of one direction of 19.27 2095. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4071.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

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

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem r19.27v

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 ⊢ (𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
2	1	ralrimiv 2965	. . 3 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜓)
3	2	anim2i 593	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
4		r19.26 3064	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	3, 4	sylibr 224	1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  r19.28v  3071  txlm  21451  tx1stc  21453  spanuni  28403