Theorem r19.28av 2568

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 2567	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ 𝜑) → ∀𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
2		ancom 264	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ 𝜑))
3		ancom 264	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
4	3	ralbii 2441	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
5	1, 2, 4	3imtr4i 200	1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wral 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-ral 2421

This theorem is referenced by:  rr19.28v  2824  fununi  5191