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Theorem r19.28v 3253
Description: Restricted quantifier version of one direction of 19.28 2263. (The other direction holds when 𝐴 is nonempty, see r19.28zv 4260.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.28v ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.28v
StepHypRef Expression
1 r19.27v 3252 . 2 ((∀𝑥𝐴 𝜓𝜑) → ∀𝑥𝐴 (𝜓𝜑))
2 ancom 453 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜓𝜑))
3 ancom 453 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43ralbii 3162 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜓𝜑))
51, 2, 43imtr4i 284 1 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wral 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006
This theorem depends on definitions:  df-bi 199  df-an 386  df-ral 3095
This theorem is referenced by:  rr19.28v  3537  fununi  6176  txlm  21779
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