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Theorem rr19.3v 3660
 Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4443 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 264 . . . 4 (𝑦 = 𝑥 → (𝜑𝜑))
21rspcv 3617 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
32ralimia 3158 . 2 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑥𝐴 𝜑)
4 ax-1 6 . . . 4 (𝜑 → (𝑦𝐴𝜑))
54ralrimiv 3181 . . 3 (𝜑 → ∀𝑦𝐴 𝜑)
65ralimi 3160 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑)
73, 6impbii 211 1 (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∈ wcel 2110  ∀wral 3138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893  df-ral 3143 This theorem is referenced by:  ispos2  17557
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