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Theorem r19.12sn 4725
Description: Special case of r19.12 3312 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 3869 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2 rexsns 4674 . 2 (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
3 rexsns 4674 . . 3 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
43ralbii 3094 . 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 205  wcel 2107  wral 3062  wrex 3071  [wsbc 3778  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-sbc 3779  df-sn 4630
This theorem is referenced by:  intimasn  42408
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