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Theorem r19.12sn 3464
Description: Special case of r19.12 2439 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
r19.12sn.1 𝐴 ∈ V
Assertion
Ref Expression
r19.12sn (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem r19.12sn
StepHypRef Expression
1 r19.12sn.1 . 2 𝐴 ∈ V
2 sbcralg 2864 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
3 rexsnsOLD 3438 . . 3 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑))
4 rexsnsOLD 3438 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
54ralbidv 2343 . . 3 (𝐴 ∈ V → (∀𝑦𝐵𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
62, 3, 53bitr4d 213 . 2 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
71, 6ax-mp 7 1 (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wcel 1409  wral 2323  wrex 2324  Vcvv 2574  [wsbc 2787  {csn 3403
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-rex 2329  df-v 2576  df-sbc 2788  df-sn 3409
This theorem is referenced by: (None)
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