Theorem r19.27av 2492

Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

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

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		ax-1 5	. . . 4 ⊢ (𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
2	1	ralrimiv 2433	. . 3 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜓)
3	2	anim2i 334	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
4		r19.26 2485	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	3, 4	sylibr 132	1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by:  r19.28av  2493  fimaxre2  10109