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Theorem rr19.3v 3339
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4055 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 252 . . . 4 (𝑦 = 𝑥 → (𝜑𝜑))
21rspcv 3300 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
32ralimia 2947 . 2 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑥𝐴 𝜑)
4 ax-1 6 . . . 4 (𝜑 → (𝑦𝐴𝜑))
54ralrimiv 2962 . . 3 (𝜑 → ∀𝑦𝐴 𝜑)
65ralimi 2949 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑)
73, 6impbii 199 1 (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1988  wral 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197
This theorem is referenced by:  ispos2  16929
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