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Theorem rr19.3v 3314
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4016 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
Assertion
Ref Expression
rr19.3v (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rr19.3v
StepHypRef Expression
1 biidd 251 . . . 4 (𝑦 = 𝑥 → (𝜑𝜑))
21rspcv 3278 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
32ralimia 2934 . 2 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑥𝐴 𝜑)
4 ax-1 6 . . . 4 (𝜑 → (𝑦𝐴𝜑))
54ralrimiv 2948 . . 3 (𝜑 → ∀𝑦𝐴 𝜑)
65ralimi 2936 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑)
73, 6impbii 198 1 (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175
This theorem is referenced by:  ispos2  16771
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