MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.12sn Structured version   Visualization version   GIF version

Theorem r19.12sn 4569
Description: Special case of r19.12 3287 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 3791 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2 rexsns 4520 . 2 (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
3 rexsns 4520 . . 3 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
43ralbii 3134 . 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 2083  wral 3107  wrex 3108  [wsbc 3711  {csn 4478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-v 3442  df-sbc 3712  df-sn 4479
This theorem is referenced by:  intimasn  39508
  Copyright terms: Public domain W3C validator