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Theorem rr19.3v 3485
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4208 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 3445 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
32ralimia 3088 . 2 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑥𝐴 𝜑)
4 ax-1 6 . . . 4 (𝜑 → (𝑦𝐴𝜑))
54ralrimiv 3103 . . 3 (𝜑 → ∀𝑦𝐴 𝜑)
65ralimi 3090 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑)
73, 6impbii 199 1 (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342
This theorem is referenced by:  ispos2  17169
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