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Theorem r19.12sn 4720
Description: Special case of r19.12 3314 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 3874 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2 rexsns 4671 . 2 (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
3 rexsns 4671 . . 3 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
43ralbii 3093 . 2 (∀𝑦𝐵𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑)
51, 2, 43bitr4g 314 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wral 3061  wrex 3070  [wsbc 3788  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-sbc 3789  df-sn 4627
This theorem is referenced by:  intimasn  43670
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