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Theorem rexsnsOLD 3410
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3409 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 2787 . . 3 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
21a1i 9 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
3 df-rex 2312 . . 3 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑))
4 velsn 3392 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
54anbi1i 431 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴𝜑))
65exbii 1496 . . 3 (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
73, 6bitri 173 . 2 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
82, 7syl6rbbr 188 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  wrex 2307  [wsbc 2764  {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-sn 3381
This theorem is referenced by:  rexsng  3412  r19.12sn  3436
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