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Theorem rr19.3v 2705
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (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 165 . . . 4 (𝑦 = 𝑥 → (𝜑𝜑))
21rspcv 2669 . . 3 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
32ralimia 2399 . 2 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑥𝐴 𝜑)
4 ax-1 5 . . . 4 (𝜑 → (𝑦𝐴𝜑))
54ralrimiv 2408 . . 3 (𝜑 → ∀𝑦𝐴 𝜑)
65ralimi 2401 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑)
73, 6impbii 121 1 (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576
This theorem is referenced by: (None)
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