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Theorem r19.12sn 4648
Description: Special case of r19.12 3321 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
Assertion
Ref Expression
r19.12sn (𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem r19.12sn
StepHypRef Expression
1 sbcralg 3854 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2 rexsns 4599 . 2 (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
3 rexsns 4599 . . 3 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
43ralbii 3162 . 2 (∀𝑦𝐵𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑)
51, 2, 43bitr4g 315 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2105  wral 3135  wrex 3136  [wsbc 3769  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-sn 4558
This theorem is referenced by:  intimasn  39880
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