Theorem r19.28v 2534

Description: Restricted quantifier version of one direction of 19.28 1525. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3421.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)

Assertion

Ref	Expression
r19.28v	⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem r19.28v

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝜑 → 𝜑)
2	1	ralrimivw 2480	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3	2	anim1i 336	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
4		r19.26 2532	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	3, 4	sylibr 133	1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wral 2390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2395

This theorem is referenced by:  txlm  12290