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Theorem r19.28av 2602
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 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.28av
StepHypRef Expression
1 r19.27av 2601 . 2 ((∀𝑥𝐴 𝜓𝜑) → ∀𝑥𝐴 (𝜓𝜑))
2 ancom 264 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜓𝜑))
3 ancom 264 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43ralbii 2472 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜓𝜑))
51, 2, 43imtr4i 200 1 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2444
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 1435  ax-gen 1437  ax-4 1498  ax-17 1514
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-ral 2449
This theorem is referenced by:  rr19.28v  2866  fununi  5256
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