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Theorem rexsnsOLD 3435
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3434 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
rexsnsOLD (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexsnsOLD
StepHypRef Expression
1 sbc5 2807 . . 3 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
21a1i 9 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
3 df-rex 2327 . . 3 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑))
4 velsn 3417 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
54anbi1i 439 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴𝜑))
65exbii 1510 . . 3 (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
73, 6bitri 177 . 2 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
82, 7syl6rbbr 192 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  wrex 2322  [wsbc 2784  {csn 3400
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 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574  df-sbc 2785  df-sn 3406
This theorem is referenced by:  rexsng  3437  r19.12sn  3461
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